Decompositions of inﬁnite graphs: Part II circuit decompositions · 2017-02-23 · We...

Journal of Combinatorial Theory, Series B 94 (2005) 278–333www.elsevier.com/locate/jctb

Decompositions of infinite graphs: Part II circuitdecompositions

François LavioletteDépartement d’informatique et de génie logiciel,Université Laval, Pavillon Adrien-Pouliot, Sainte-Foy,

Québec, Canada G1K 7P4

Received 28 May 1997Available online 5 March 2005

Abstract

We characterize the graphs that admit a decomposition into circuits, i.e. finite or infinite connected2-regular graphs. Moreover, we show that, as is the case for the removal of a closed eulerian subgraphfrom a finite graph, removal of a non-dominated eulerian subgraph from a (finite or infinite) graphdoes not change its circuit-decomposability or circuit-indecomposability. For cycle-decomposablegraphs, we show that in any end which contains at leastn+ 1 pairwise edge-disjoint rays, there arenedge-disjoint rays that can be removed from the graph without altering its cycle-decomposability.Wealso generalize the notion of the parity of the degree of a vertex to vertices of infinite degree, and inthis way extend the well-known result that eulerian finite graphs are circuit-decomposable to graphsof arbitrary cardinality.© 2005 Elsevier Inc. All rights reserved.

Keywords:Infinite graphs; Eulerian graph; Decomposition; Circuits; Cutset

1. Introduction

As is well known, the subject of eulerian graphs originates from the Königsberg bridgeproblem that was solved in 1736 by Euler. In the finite case, the principal theorem on thistopic (due in part to Euler, Hierholzer and Veblen), says in substance that for a connectedgraph, beingedge-traceable (i.e., havinga closedeulerian trail), havingadecomposition intoconnected 2-regular subgraphs (circuits) and being eulerian are equivalent properties. In theinfinite case, these three properties are no longer equivalent. It is known that for a finite or

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F. Laviolette / Journal of Combinatorial Theory, Series B 94 (2005) 278–333 279

infinite graph, to be eulerian (i.e., not to contain any odd vertices) is equivalent to admittinga decomposition into edge-traceable graphs.Also, Nash-Williams has shown that the graphswhich admit a decomposition intofiniteconnected 2-regular subgraphs (cycles) are exactlythe ones that contain no edge-cut of odd cardinality. The present paper characterizes thegraphs that admit a decomposition into circuits (i.e. finite or infinite connected 2-regulargraphs).In contrast to the two characterizations just mentioned, no characterization of the graphs

that are circuit-decomposable may be based only on the degrees of the vertices or on thecardinality of the edge-cuts. Indeed, there exist graphsG andH which differ by a singleedge joining two vertices which cannot be separated by a finite cut in either, such thatG isdecomposable whereasH is not. (For example, take the graph of Fig.4, Section 4 to beHand the same graph minus the edge joining the two vertices of infinite degree forG.)We approach the problem of the characterization in two ways.Our first approach is to introduce a notion of local circuit-decomposability for certain

specifiedparts of agraph, and then to show that if thosespecifiedparts of thegraphare locallycircuit-decomposable then the graph is circuit-decomposable in its entirety. The interestingparts of the graph in this approach are calledregions: they are connection-induced subgraphsthat are joined to the rest of the graph by only a finite (even or odd) number of edges. Regionsare said to be even or odd according as this number of edges is even or odd. Among all theregions, theperipheralones, which are odd regions not containing any odd cut of the graph,play a key role in this approach since it turns out that a graph is circuit-decomposable if andonly if its peripheral regions are locally circuit-decomposable (Theorem 7.5).Hence the peripheral regions are the parts of the graphwhere onemay expect to encounter

serious difficulties in connection with circuit decomposition. At the other extreme, theregions that are most easily handled are those that are locally cycle-decomposable; byNash-Williams’s Theorem, such regions must be even and, like peripheral regions, cannotcontain odd cuts. If one considers the two types of regions (peripheral and locally cycle-decomposable) in the case of finite graphs, one easily sees that they have the followingtwo properties: (1) peripheral regions always contain at least one odd vertex; (2) locallycycle-decomposable regions contain even vertices only. For infinite graphs, this is no longertrue because of the vertices of infinite degree that may be considered to be both of evenand odd degree. This leads us to a generalization of the parity of the degree of a vertex towhat we call theparity type. This is done in such a way that properties (1) and (2) extendto infinite graphs, and that the following conditions are satisfied:(3) any two vertices in the same�-class (i.e., which are connected by infinitely many

pairwise edge-disjoint paths) have equal parity type;(4) as for the parity of the degree, the removal from the graph of a finite eulerian subgraph

does not alter the parity type of any vertex.With this definition we show that having noodd-typevertex is a sufficient condition fora graph to admit a circuit decomposition, whereas having noodd vertex is a necessarycondition. In particular, this gives the result that an infinite vertex in a transitive graph isalways of even type, and hence for these graphs to be eulerian is a necessary and sufficientcondition for the existence of a circuit decomposition.Furthermore, a peripheral region must always contain an odd number of�-classes of

odd-type vertices. This corresponds to the situation in the finite case where every peripheral

280 F. Laviolette / Journal of Combinatorial Theory, Series B 94 (2005) 278–333

region contains an odd number of odd vertices (note that in a finite graph the�-classes aresingletons). Moreover, if a peripheral regionA of a graphG contains exactly one odd-type�-class, thenA is always locally cycle-decomposableto within a single edge, even if it isnot locally circuit-decomposable. Hence, to study the circuit-decomposability of a graph,we can restrict ourselves to regions which are “almost” cycle-decomposable and thereforehave a much simpler structure than the whole graph.Odd-type and even-type vertices also play an important role in our second approach to

the problem. This approach is based on the ray structure of the graph: we say that a graphGhasenough raysif any odd region ofG contains a ray. It is easy to see that having enoughrays is a necessary condition for a graph to admit a circuit decomposition. On the otherhand—relativizing the definition in the obvious way—we show that having enough non-dominated rays is a sufficient condition (Theorem6.9); in fact, we show that it is a necessaryand sufficient condition for a graph to admit a decomposition intonon-dominatedcircuits.This leads us to ask whether having enough rays with some specified property might bea necessary and sufficient condition for a graph to admit a circuit decomposition. Indeed,such a property exists and will be calledeligibility; to describe it we need the followingobservations.It follows from Theorem 6.9 that, as for the eulerian subgraphs of a finite graph, the

removal of an eulerian non-dominated subgraphH from a graphGdoes not affect its circuit-decomposability or circuit-indecomposability (seeTheorem 8.1). Defining aneulerian-typegraph as a graph that contains no odd-type vertex, we further show (Theorem 8.6) thatthis property also holds ifH is a dominated eulerian-type subgraph provided that noneof its dominating vertices is of odd type in the graphG\H . In both situations we speakof removablesubgraphs. In terms of this definition, we say that a rayR is eligible if itis contained in an “almost eulerian” locally finite removable graph. Any eligible ray of acircuit-decomposable graph is always contained in a 2-ray (a graph that is the edge-disjointunion of two rays having the same origin) that is a member of a decomposition ofG intocycles and 2-rays. Since any decomposition into cycles and 2-rays trivially induces a circuitdecomposition, we can see eligible rays as the rays that can be used if one want to constructa circuit decomposition. Moreover, our main theorem (Theorem 9.5) says that a graphGhas a circuit decomposition if and only if it has enough eligible rays.Eligible rays exist in most ends (i.e., classes of finitely inseparable rays). More precisely,

eligible rays occur in every non-dominatedend, in every end that contains at least threeedge-disjoint rays (i.e. is of�-multiplicity �3), and also in every end that is of�-multiplicity �2and is dominated by some odd-type vertex.The preceding results show that a non-circuit-decomposable graph must contain an odd

region whose ends are very “thin”. Further, these graphs must contain an odd region whichis of one of the four types that are shown in Fig. 9 (Section 9) or a suitable combination ofthem.

2. Definitions and preliminaries

In this section we introduce basic definitions and present results that will be needed in theproofs leading up to our main theorem. Most of these results are known, but some are new.


The latter have been grouped together in this section because they are also of independentinterest.

2.1. Generalities

For the purposes of this paper, we assume all graphs to be unoriented, without loops ormultiple edges, unless otherwise stated (multiple edges are allowed in quotient graphs).The symbolG will always denote a graph. Aneuleriangraph is a graph (not necessarilyconnected) whose vertices are all of even or infinite degree.Pathsare understood to befinite. ForX, Y ⊆ V (G) anXY-pathis a path whose endpoints are inXandY, respectively;whenX orY is a singleton, we omit curly brackets. IfS, T are subgraphs ofG, anST-pathis aV (S)V (T )-path. We shall speak of anX-pathor anS-pathin place of anXX-path or anSS-path, respectively. Aray is a connected graph having exactly one vertex of degree one,called theorigin of the ray, and all the others of degree 2. GivenX ⊆ V (G), anX-ray is aray having its origin inX. As in the case of paths we shall also speak ofu-rays andS-rays,whereu ∈ V (G) andS is a subgraph ofG. A set of� edge-disjoint rays(Ri)i∈I , where�is some cardinal, is said to be a�-tresseif

⋂i∈I V (Ri) is an infinite set.

A circuit is a non-empty connected 2-regular graph. A finite circuit is called acycleandan infinite one, adouble-ray. A tail of a ray or double-rayR is a subray ofR. A trail isa sequence of consecutively adjacent vertices such that the edges joining two consecutivevertices of the sequence are all distinct.A trail may be finite, 1-way infinite or 2-way infinite.Whenever convenient the word trail will also be applied to the graph formed by the verticesand edges of the sequence. Anedge-traceablegraph is a graph obtained from a circuit byidentifying some of its vertices but no edges and that contains no loop and nomultiple edge.In other words, an edge-traceable graph is a graph which is either a finite eulerian trail ora 2-way infinite (and hence eulerian) trail. The finite edge-traceable graphs are thereforeexactly the connected eulerian ones; for the infinite edge-traceable graphs, i.e., the ones thatare a 2-way infinite trail, Erd˝os et al.[3] give the following characterization:

Theorem 2.1. A connected graph G is a2-way infinite trail if and only if

(i) E(G) is countably infinite;(ii) G is eulerian;(iii) there is no finite set of edges whose deletion leaves more than two infinite components;

and(iv) there is no finite eulerian subgraph the deletion of whose edges leaves more than one

infinite component.

Note that Erd˝os et al.[3] also give a similar characterization of graphs that are a 1-wayinfinite trail (i.e., graphs obtained from a ray by identifying some of its vertices but noedges).Given two subgraphsA,B of a graphG, we denote by[A,B]G the set of edges ofG

that have one endpoint inV (A) and the other inV (B), andB − A denotes the inducedsubgraph ofGonV (B)−V (A). By abuse of language, we will frequently identify a singlevertex with the graph that consists of this vertex only, and similarly a single edge may be


identified with the graph that consists only of this edge and its two incident vertices. Thusin the preceding definitions,A can also be a vertex or a single edge. Acutof a graphG isa set of edges of the form[A,G− A]G. If no confusion is likely, we shall writeA insteadof G − A, and, unless otherwise stated, bothA andB will be inducedsubgraphs ofG. Aregionof a graphG is a connected non-empty induced subgraphAsuch that[A,A]G is finite(possibly empty);A is anevenor odd regionaccording as

∣∣[A,A]G∣∣ is even or odd. Abond

is a non-empty cut which is minimal with respect to inclusion. Observe that a cut[A,A]Gof a connected graph is a bond if and only if bothA andA are connected. In general, everycut [A,A]G is a union of edge-disjoint bonds (this is well known in the finite case; for aproof in the general case, see[11, Remark 1]). Hence, in the case where[A,A]G is finite,each componentK ofA is a region ofG since[K,K]G ⊆ [A,A]G. The following lemma isuseful in showing how one can construct finite cuts by transferring vertices from one sideof a finite cut to the other.

Lemma 2.2. Let [A,A]G and[B,B]G be two finite cuts of G. Then[A−B,A− B]G and[A ∩ B,A ∩ B]G are also finite cuts, and if, moreover, [B,B]G is even andB ⊆ A, then[A,A]G and[A− B,A− B]G are either both even or both odd.

Proof. The first assertion follows from the fact that both[A − B,A− B]G and [A ∩B,A ∩ B]G are contained in[A,A]G ∪ [B,B]G. For the second assertion, note that whenB ⊆ A, we have

∣∣[A− B,A− B]G∣∣= ∣∣[A,A]G

∣∣− ∣∣[B,A]G∣∣+ ∣∣[B,A− B]G

∣∣= ∣∣[A,A]G

∣∣− ∣∣[B,A]G∣∣+ ∣∣[B,B]G

∣∣− ∣∣[B,A]G∣∣

= ∣∣[A,A]G∣∣− 2

∣∣[B,A]G∣∣+ ∣∣[B,B]G

∣∣. �

If A is a set of edges,G\A is the graph obtained by the removal of all edges ofA (retainingall vertices). For a non-empty graphH,G\H denotes the graphG\E(H). BdryG(H) is theset of all vertices ofH that are incident with an edge ofG\H . G/H is the quotient graphobtained by identifying all the vertices ofH and by removing all the loops thus obtained;G/H may have multiple edges. We extend the definition of a quotient graphA/H to thecase whereA is a subgraph ofG by takingA/H := A/(A∩H). There will be situations inwhich the setV (H) appears both as a set of vertices ofG and as a vertex ofG/H ; to avoidconfusion we shall denoteV (H) by q

Hwhen it is a vertex of the quotient.

Two verticesxandyare said to beinfinitely edge-connectedif there exist infinitely manypairwise edge-disjointxy-paths or equivalently if the two vertices cannot be separated by afinite cut ofG. This is an equivalence relation onV (G); its classes are called�-classes.The next lemma shows that it is possible to transfer an�-class (possibly with some other

vertices) from one side of a finite cut to the other and preserve the finiteness of the cut.

Lemma 2.3. Let A be an induced subgraph of G such that[A,A]G is finite, x ∈ V (A) andY be a finite set of vertices that is disjoint from the�-class of x. Then there exists a regionB ⊆ A that contains x(and hence its�-class) but no vertex of Y.


H4

H3

H2

H1

H0

Fig. 1.A graph having a non-dominated�-dominated end. The rays contained inH3\H4 do not belong to that end.

Proof. For eachy ∈ Y , fix a finite cut[By, By]G such thaty ∈ V (By) andx ∈ V (By)and putC := A ∩ ⋂

y∈Y By . Then [C,C]G is finite since it is contained in[A,A]G ∪⋃y∈Y [By, By]G. Therefore the component ofC that containsx is the desired region.�

2.2. Dominated rays and end-equivalent rays

A rayR is said to bedominated(resp.�-dominated) by a vertexx in a graphG if for anyfinite setS ⊆ V (G) − {x} (resp.S ⊆ E(G)), some tail ofR lies in the same componentof G − S (resp.G\S) asx or, equivalently, if there exist infinitely manyxR-paths ofGwhich pairwise intersect inxonly (resp. which are pairwise edge-disjoint and have differentendpoints inR). By abuse of language we will frequently omit mention of the dominatingvertex and simply speak of dominated rays. IfH is a subgraph ofG the statement “a rayR isdominated inH ” will mean thatR is contained inH, and is dominated inH by a vertex ofH.The same convention will be applied to�-domination. Note that a ray which is dominated(resp.�-dominated) inH is still dominated (resp.�-dominated) inG, whereas the converseis not necessarily true. However, ifH is a region, then it is easy to see that a ray ofH isdominated (resp.�-dominated) inH if and only if it is dominated (resp.�-dominated) inG.For this reason, we shall not state explicitly whether a ray of a region is dominated (resp.�-dominated) inG or in the region. It is obvious that any vertex which dominates some rayin a graph also�-dominates it; however, as is shown in Proposition2.19 and by Fig. 1, theconverse is not true. Further, the set of all the vertices which�-dominate some ray is alwaysan�-class of the graph, provided it is non-empty; but this is not true of the vertices whichdominate a ray.Two raysR1, R2 areend-equivalentin a graphG (in symbols,R1 ∼ R2) if for every

finite subsetS of V (G), some tails ofR1 andR2 lie in the same component ofG − S

or, equivalently, if there exist infinitely many (vertex-)disjointR1R2-paths ofG. It is wellknown that∼ is indeed an equivalence relation. The equivalence classes are calledends(see Diestel [1] for a survey). Clearly, a�-tresse is always contained in a single end. Givena set of endsR, we will abuse language and say that a ray belongs toR if it belongs to anend ofR. An end or a�-tresse� is said to bedominated(resp.�-dominated) by a vertexx in G if x dominates (resp.�-dominates) some (and therefore all) rays in�; again we will


frequently omit mention of the dominating vertex. The�-multiplicity of an end� is themaximal number of edge-disjoint rays of�, or to be more precise:

�-multiplicity of � := sup{|�| : � is a set of edge-disjoint elements of�}.As in the case of themultiplicity defined byHalin[8] (supremumof the number of vertex-

disjoint rays in�), the supremum for the number of edge-disjoint rays is actually attained.The proof is similar to that of Halin’s [8] Satz 1, which is itself relies on Halin’s [7] Satz 1.

Remark 2.4. An end� that has infinitely many dominating vertices always contains an�-tresse, and hence has infinite�-multiplicity. One can easily construct an�-tresse in�in the following way. LetD be the set of vertices that dominate� in G, and letx0 ∈ D.Let P 0

0 be any non-degeneratex0D-path ofG, andx1 be the end-vertex ofP 00 that is not

x0. Let P 10 be any non-degeneratex1D-path ofG − (P 0

0 − {x1}), x2 be the end-vertex ofP 10 that is notx1, andP 0

1 be anyx0x1-path ofG − (P 00 ∪ P 1

0 − {x0, x1}). Let P 20 be any

non-degeneratex2D-path ofG− (P 00 ∪ P 1

0 ∪ P 01 − {x2}), x3 be the end-vertex ofP 2

0 thatis notx2, P 1

1 be anyx1x2-path ofG − (P 00 ∪ P 1

0 ∪ P 01 ∪ P 2

0 − {x1, x2}), andP 02 be any

x0x1-path ofG− (P 00 ∪ P 1

0 ∪ P 01 ∪ P 2

0 ∪ P 11 − {x0, x1}). Continue this construction, and

putRn := ⋃i∈� P

in to obtain the desired family.

One of the best-known results on ends is Halin’s theorem[6] on the existence of anend-faithful spanning tree for countable connected graphs.

Theorem 2.5. Every connected countable graph G contains an end-faithful spanning tree,i.e., a spanning tree T such that

(1) no two disjoint rays of T are end-equivalent in G;(2) every ray of G is end-equivalent in G to some ray of T

A particularly interesting case of end-faithful spanning tree is the normal tree.

Definition 2.6. Aspanning treeTwith rootr of a graphG is callednormalif the endverticesof every edge ofG are comparable in the natural tree order onV (G) induced byT.

Normal spanning trees are known in the finite case as depth-first search trees, and it is adirect consequence of the following lemma that they always are end-faithful.

Lemma 2.7(Diestel and Leader[2] ). Let T be a normal tree of a graph G. Then every rayin G meets some ray of T infinitely often.

The next theorem characterizes the graphs that admit normal trees. Since a finite set ofvertices is always a dispersed set (see below for the definition), it follows directly from thattheorem that normal trees always exist in countable graphs.

Theorem 2.8(Jung[9] ). A connected graph G has a normal spanning tree if and only ifV (G) is a countable union of dispersed sets,where a dispersed set is a set of vertices suchthat every ray is separable from it by a finite number of vertices of G.


The last two results imply the following corollary that we will need latter on.

Corollary 2.9. Any end� of a countable graph contains a ray that meets any other ray of� infinitely often.

An end-faithful spanning tree is often viewed as a representative of the ends of the originalgraph. The next result is about spanning trees that represent no end at all.

Theorem 2.10(Polat [14] and Širán [16] ). A connected countable graph has a raylessspanning tree if and only if every ray in G is dominated.

The next three propositions show how domination and end-equivalence are linked.

Proposition 2.11 (Laviolette and Polat[12] or Hahn et al. [5] for a more direct proof).Given an infinite set of vertices X of a connected graph G, there exist a vertexx ∈ V (G)and an infinite family of xX-paths which pairwise intersect in x only, or a ray Q and aninfinite family of vertex-disjoint QX-paths.

For our purposes, the following consequence of this proposition is more relevant.

Proposition 2.12. Let R be any ray of G. If a connected subgraph A ofG\R containsinfinitely many vertices of R, then there exists a vertexx ∈ V (A) that dominates R inA∪R,or a rayQ ⊆ A that is end-equivalent to R inA ∪ R.

Proposition 2.13. Let R, R′ be two end-inequivalent rays of G, andx ∈ V (G). If x domi-nates R in G then x will dominate a tail of R in bothG− (R′ − x) andG\R′.

Proof. SinceG − (R′ − x) is a subgraph ofG\R′, we only have to show the result forG− (R′ − x). LetF be an infinite set ofxR-paths ofGwhich pairwise intersects inx only.If x does not dominateR inG− (R′ − x), thenR′ − x must meet all but a finite number ofpaths ofF . Since each such path ofF contains anRR′-path disjoint fromx, RandR′ aretherefore end-equivalent inG, a contradiction. �

Definition 2.14. A family (Hn)n∈� of connected subgraphs ofG is said to be astratifyingsequenceif Bdry(Hn) is a non-empty finite set andHn+1 ⊆ Hn − Bdry(Hn) for anyn. If,moreover,[Hn,Hn]G is finite for anyn (i.e., if eachHn is a region), we then speak of an�-stratifying sequence.

Remark 2.15. Given a stratifying sequence(Hn)n∈�, it follows from the second propertyof the definition that a vertex inHn must be at distance at leastn from any vertex ofBdry(H0). Therefore

⋂n∈�Hn = ∅.

Remark 2.16. If a ray Rmeets infinitely many members of a stratifying sequence theneach member of that sequence contains a tail ofR.


The next proposition shows that there is a strong link between stratifying sequences andnon-dominated ends.

Proposition 2.17.

(i) A ray of a graph G is not dominated(resp. not�-dominated) in G if and only if it meetsevery member of some stratifying(resp.�-stratifying) sequence;

(ii) the set of all the rays thatmeet eachmember of a stratifying(resp.�-stratifying) sequenceis a non-dominated(resp. non-�-dominated) end.

Proof. We shall give a proof for the dominated case; the�-dominated case is very similarand left to the reader.(i) Necessity. SupposeR is not dominated inG and observe that for any finite set of

verticesS, there exists another finite set of verticesX ⊆ V (G) − S such that the uniquecomponent ofG − X that contains a tail ofR does not meetS. Otherwise, by Menger’sTheorem, there exists an infinite family ofSR-paths any two of which are either disjoint ormeet on their end-vertex inSonly. Hence somes ∈ S must be an end-vertex of an infinitenumber of thoseSR-paths, contradicting the hypothesis thatR is not dominated.We will now construct two sequences(Xn)n∈� and(Hn)n∈�, consisting of finite subsets

of V (G) and connected subgraphs ofG, respectively. Letx be any vertex belonging to thecomponent ofG that containsR, andX0 be a finite set of vertices ofV (G) − x such thatthe unique component ofG − X0 that contains a tail ofR does not containx. LetH0 bethat component. Then letX1 be a finite set of vertices ofV (G)− (X0 ∪ {x}) such that thecomponent ofG − X1 that contains a tail ofR does not meetX0 ∪ {x}. Let H1 be thatcomponent. Repeat this construction mutatis mutandis to obtain the sequence(Hn)n∈�.Clearly the sequence is stratifying since eachHn has its boundary contained in the finitesetXn.Sufficiency. By Remark2.16, eachHn contains a tail ofR. Bdry(Hn) being finite, any

vertex that dominatesR in Gmust belong toHn, for all n, a contradiction to Remark 2.15.(ii) Let (Hn)n∈� be a stratifying sequence ofG and� the set of all rays that meet each

Hn. SinceX := ⋃n∈� Bdry(Hn) is an infinite set of vertices, it follows from Proposition

2.11 that there exist a vertexx ∈ V (G) and an infinite family ofxX-paths which pairwiseintersect inxonly, or a rayQand an infinite family of vertex-disjointQX-paths. In this casethe first possibility cannot occur because such anx has to belong to infinitely manyHn’s,and therefore to all of them, contradicting Remark 2.15. Since any such rayQmust clearlybelong to�, the latter is therefore non-empty, and it is then easy to see that� is an end.Moreover,� is a non-dominated end since, by (i), no ray of� is dominated. �

Corollary 2.18. Let H be a connected subgraph ofG all of whose rays are dominated(resp.�-dominated) in G. Then, for any stratifying(resp.�-stratifying) sequence(Hn)n∈� of G,there existsn0 ∈ � such thatV (H) ∩ V (Hn) = ∅ for anyn�n0.

Proof. TheHn’s being stratifying, clearly ifV (H)∩V (Hn) is empty for somen = n0, thenit is alsoempty for anyn�n0.Hence,bywayof contradictionsuppose thatV (H)∩V (Hn) �=∅ for all n. LetT be a spanning tree ofH andx ∈ V (T ) (= V (H)). By Remark2.15, there


is an indexn1 such thatx �∈ V (Hn) for any n�n1, and letT1 be the union of allxy-paths ofT with y ∈ ⋃

n�n1 BdryG(Hn). ClearlyT1 is infinite but sinceT1 is a subtreeof T and theHn’s are stratifying,T1 − Hn is finite for anyn�n1. ThusT1 is an infinitelocally finite connected graph, and therefore contains anx-ray R (say). It follows fromthe finiteness of eachT1 − Hn thatRmeets eachHn for any n�n1 (and hence for anyn). Thus, by Proposition2.17(i),R is not dominated (resp.�-dominated) inG, contrary tohypothesis. �

The next proposition says that an end that is�-dominated but not dominated must be verysimilar in structure to the unique end of the graph in Fig. 1.

Proposition 2.19. Let � be a set of rays of G and U an�-class of G. Then the followingstatements are equivalent:

(i) � is a non-dominated end that is�-dominated by some(or equivalently, each) vertex ofU;

(ii) There exists a stratifying sequence(Hn)n∈� such that U meets eachHn, and� is theset of all rays that have a tail in eachHn;

(iii) � is anon-dominatedend that containsan�-tresse(Ri)i∈� such that⋂i∈� V (Ri) ⊆ U .

Proof. Without loss of generality, we may supposeG to be connected, because otherwisewe may consider only the component ofG that contains the�-classU.(i) ⇒ (ii ). Let R ∈ � and let(Hn)n∈� be a stratifying sequence given by Proposi-

tion 2.17(i). By Remark 2.16,Rhas a tail in eachHn, and clearly a ray is end-equivalent toR in G if and only if it has the same property. Hence� is the set of all rays that have a tailin eachHn. Now, by way of contradiction, suppose thatV (Hn) ∩ U = ∅ for somen ∈ �.Let Rn be a tail ofR contained inHn andu be an element ofU. Sinceu �-dominatesR,there exist infinitely many edge-disjointuRn-paths. Sinceu ∈ U ⊆ V (G)− V (Hn), eachof these paths includes an element of Bdry(Hn). Since Bdry(Hn) is finite, it follows thatuis connected to somex ∈ Bdry(Hn) by infinitely many edge-disjoint paths, and sox ∈ U ,contradicting the assumption thatV (Hn) ∩ U = ∅.(ii )⇒ (iii ). It follows fromProposition 2.17(ii) that� is a non-dominated end. Letu ∈ U .

By Remark 2.15, there is an indexn0 such thatu �∈ V (Hn) for all n�n0. By truncating thesequence(Hn)n∈�, we may therefore suppose thatu �∈ V (H0). �

Claim. There exists anX ⊆ ⋃n∈� Bdry(Hn), and a connected subgraph T of G such that

(1) X meets eachBdry(Hn) (and hence is infinite);(2) T contains u and X, and anyx ∈ X is infinitely edge-connected to u in T;(3) given any two distinct vertices x, x′ of X, either x separatesx′ from u in T or vice versa.

If we suppose the claim to be true then by (3),X is a set of cut-vertices ofT, and moreover,X is strung out along a ray in the block-cutpoint tree ofT. Hence, by (2),T contains aninfinite family of edge-disjointu-raysRi , i ∈ �, each containingX. Clearly theRi ’s forman�-tresse, and by (1) eachRi belongs to�. Moreover,

⋂i∈� V (Ri) being contained in the

�-class ofT containingu, it follows that⋂i∈� V (Ri) ⊆ U (the�-class ofG containing

u), and we are done.


Proof. Let R ∈ �. SinceU is an�-class that meets eachHn, and since theHn’s areconnected,u �-dominatesR in G.Choose an enumeration{x1, x2, . . .} of ⋃n∈� Bdry(Hn). Starting withG0 = G, we

inductively define a decreasing sequence of subgraphsGi of G such that

(a) xi+1 ∈ V (Gi), andGi+1 is defined fromGi either by the deletion of at most one vertex(viz.xi+1) or the deletion of a finite number of edges (hence eachGi contains a tail ofR);

(b) all tails ofRcontained inGi are�-dominated inGi by u;(c) if xi ∈ V (Gi) then it is a cut-vertex ofGi that separatesu in Gi from some tail ofR.

SupposeGi has already beendefined. In viewof (c), wemake the following case distinction:Case1: xi+1 is a cut-vertex ofGi which separatesu from some tail ofR. In this case put

Gi+1 := Gi .Case2: Some componentK ofGi −xi+1 contains bothuand a tail ofR. If u �-dominates

some tail ofR in K, takeGi+1 := Gi − xi+1. If u does not�-dominate any tail ofRin K, then there exists a finite bond[A,A]K that separatesu from some tail ofR. PutGi+1 := Gi\[A,A]K .It is easy to see that theGi ’s satisfy conditions (a)–(c), and that at most a finite number

of Hn’s are not subgraphs ofGi for any i.Let T be the component of

⋂i∈�Gi that containsu andX := {xi : xi ∈ V (T )}, and

let us show they have the claimed properties. It is easy to see thatxi+1 �∈ X if and only ifGi+1 = Gi − xi+1. This implies that Condition (1) is satisfied for anyn because otherwiseGi+1 = Gi − xi+1 for everyxi+1 ∈ BdryG(Hn), implying that there exists a finite numberj such that BdryG(Hn) ∩ V (Gj ) = ∅, contradicting (b) fori = j .It follows from (b) and (c) that any vertexxi ∈ X is infinitely edge-connected tou inGi ,

and hence, again because of (b), in eachGj with j� i. Moreover it follows from (c) thatxiseparatesu from someHm. If im is the smallest index such thatxj ∈ V (Hm) for anyj� im,then everyuxi-path ofGim is contained inT. Condition (2) is therefore satisfied.Let xk, xl (k < l) be any two vertices ofX. SinceGl ⊆ Gk and because of (b) and (c),

in the graphGl , eitherxk separatesxl from u or xl separatesxk from u. Thus Condition (3)is satisfied becauseT ⊆ Gl and because it follows from Condition (2) thatu, xk andxl allbelong toT.(iii )⇒ (i). Clearly� is �-dominated by each vertex of the infinite subset

⋂i∈� V (Ri) of

U. Hence� is �-dominated by each vertex ofU. �

Corollary 2.20. A non-dominated�-dominated end has infinite�-multiplicity.

Corollary 2.21. An �-dominated end that contains no�-tresse has a finite and strictlypositive number of dominating vertices.

Proof. Remark2.4 and Proposition 2.19.�

Among all the rays that are�-dominated by a given vertexu, some seem “closer” touthan others; a closer ray being “in themiddle” betweenuand the ray farther away. Formally,given two edge-disjoint raysQ andQ′ that are�-dominated byu, we say thatQ is closer tou thanQ′ in G (denoted byQ ≺u Q′) if Q′ is not�-dominated byu in G\Q.


It follows from the definition of�-domination thatQ ≺u Q′ if and only if given anyinfinite family F of edge-disjointuQ′-paths having distinct end-vertices onQ′, at mostfinitely many members ofF are edge-disjoint fromQ. Thus, it is easy to see that≺u is atransitive relation. Moreover, since the paths ofF can be chosen to be edge-disjoint fromQ′, Q ≺u Q′ implies thatQ′ �≺u Q. Hence�u is a partial order. The following resultshows that, under very mild conditions,≺u has a certain Noetherian property.

Proposition 2.22. Let u be a vertex that does not�-dominate any�-tresse in G. Then, forany infinite ascending chainQ0 ≺u Q1 ≺u Q2 ≺u . . . of edge-disjoint rays�-dominatedby u, all but a finite number of theQi ’s are end-equivalent.

The proof is based on the following lemma.

Lemma 2.23. A vertex u�-dominates an�-tresse in G if and only if there exists an infinitesubset U of the�-class of u such that for each finite setS ⊆ V (G), all but a finite numberof vertices of U belong to the same�-class ofG− S.

Proof. WithU := ⋂i∈� V (Ri) for some�-tresse(Ri)i∈� �-dominated byu, the necessity

is an immediate consequence of the definition of�-domination. For the sufficiency, for eachfinite setS ⊆ V (G), denote byUS the only infinite set which is of the formU ∩ X forsome�-classX of G − S, and let us recursively construct a ray that is�-dominated byuin G as follows. Fixx0 ∈ U . LetW1 be anx0U{x0}-path ofG which is internally disjointfrom U{x0}, and denote byx1 the end-vertex ofW1 that belongs toU{x0}. It follows fromV (W1) ∩ U{x0} = {x1}, thatU{x0} = UV (W1−x1). Hencex1 ∈ UV (W1−x1), implying thatx1andUV (W1) lie in the same component ofG − (W1 − x1). LetW2 be anx1UV (W1)-pathof G − (W1 − x1) that is internally disjoint fromUV (W1), and denote byx2 its end-vertexcontained inUV (W1). Sincex2 ∈ UV (W1∪W2−x2), we can then choose anx2UV (W1∪W2)-pathW3 that is internally disjoint fromUV (W1∪W2), etc.Clearly,R := W1∪W2∪ · · · is a ray. Moreover, sinceRmeetsU (and hence the�-class

of u) infinitely often,R is �-dominated byu in G.Let � be the end ofR in G, andD its set of dominating vertices. IfD is infinite then,

by Remark2.4, we are done. Hence supposeD to be finite (possibly empty). Then byProposition 2.19(i)⇒ (iii ) applied toG−D, we also are done because a tail ofRcontainedin G−D is non-dominated inG−D but �-dominated by each vertex ofUD, and becauseany ray end-equivalent to a tail ofR in G−D is end-equivalent toR in G. �

Proof of Proposition 2.22.We first note that ifQj andQk (j < k) belong to the sameend� of G, then so do allQl ’s with j� l�k. Assume the contrary and choosel such thatj < l < k andQl �∈ �. SinceQl �≺u Qj , there exists an infinite familyF1 of edge-disjointuQj -paths, edge-disjoint fromQl and pairwisely having different end-vertices onQj . Moreover, sinceQl �∈ �, there also exists an infinite familyF2 of vertex-disjointQjQk-paths that are vertex-disjoint fromQl . SinceQl is edge-disjoint fromQj , one can thereforeconstruct an infinite family of edge-disjointuQk-paths contained in

⋃F1 ∪Qj ∪⋃F2.This is a contradiction toQl ≺u Qk becauseQl is edge-disjoint from

⋃F1 ∪ Qj ∪⋃F2.


Suppose that the conclusion of the proposition is false. By the preceding paragraph, noend ofG may contain an infinite number ofQi ’s. In other words, there exists an infinitesubsequence of theQi ’s that are pairwise end-inequivalent, and without loss of generality,this subsequence may be taken as the whole sequence itself.Now, let us show that ifQj andQk (j < k) are both dominated by some vertexx then

k = j + 1. By way of contradiction, supposek > j + 1. By Proposition2.13, it followsfrom the end-inequivalence of theQi ’s that a tail ofQj and a tail ofQk are both dominatedby x inG\Qj+1. Since moreoveru still �-dominatesQj inG\Qj+1, we therefore have thatfor any finite setS ⊆ E(G\Qj+1), the vertexu, a tail ofQj , the vertexx and a tail ofQklie all in the same component of(G\Qj+1)\S. Thusu still �-dominatesQk in G\Qj+1,contradicting the fact thatQj+1 ≺u Qk.Hence there is an infinite subsequence of theQi ’s no two of whose members share

a dominating vertex; without loss of generality, suppose that it is the wholesequence.EachQi has a dominating vertexxi . This follows from the fact thatQi is �-dominated

by u, and hence ifQi were non-dominated then by Proposition 2.19, the end ofQi wouldcontain an�-tresse, contradicting the hypothesis. The setU := {xi : i ∈ �} is infinite andcontained in the�-class ofu. Hence by Lemma 2.23 there exists a finite setS ⊆ V (G) suchthat for anyi, {xj : j� i} meets more than one�-class ofG− S. Suppose that among allsuch sets,Shas minimal cardinality. SinceS �= ∅, fix s ∈ S and leti0 be the smallest indexsuch that{xj : j� i0} is contained in a single�-class ofG − (S − {s}), and such that noQj (j� i0) is dominated bys in G. Since eachxj dominatesQj , s �∈ {xj : j� i0}.Let i1 and i2 be two indices withi0 < i1 < i2 such thatxi1, as a vertex ofG − S,

is neither in the�-class ofxi0 nor in the one ofxi2 (xi0 may be in the same�-classasxi2).Weclaim that thereexists an infinite familyF of edge-disjointxi0xi2-pathsofG−(S−{s})

that are edge-disjoint fromQi1. If xi0 andxi2 belong to the same�-class ofG − S, thenthe claim follows from the fact thatxi1 dominates a tail ofQi1 in G − S. In that case, thepaths of the desired family can even be chosen inG− S. If xi0 andxi2 belong to different�-classes ofG − S, then again because of the dominating property ofxi1 and becausesdoes not dominateQi1 inG, there exists inG− (S−{s}) an infinite family of edge-disjointxi0s-paths, and an infinite family of edge-disjointsxi2-paths, which both consist of pathsthat are edge-disjoint fromQi1. Clearly, in the union of all the paths in these two families,one can construct the desired familyF .To finish the proof, we will now construct an infinite family of edge-disjointuQi2-paths

which are edge-disjoint fromQi1 and have distinct end-vertices onQi2; the existence ofsuch a family being in contradiction withQi1 ≺u Qi2. LetFik (k = 0 or 2) be an infinitefamily of xikQik -paths, which pairwise intersect inxik only and are edge-disjoint fromQi1.Such families exist becausexik dominatesQik inG, andQik andQi1 are end-inequivalent,k = 0,2. Finally, sinceQi0 ≺u Qi1, one can construct an infinite familyFu of edge-disjointuQi0-paths, pairwise having distinct end-vertices onQi0, and all being edge-disjoint fromQi1.It is easy to see that inH := ⋃Fu∪⋃Fi0 ∪⋃F ∪⋃Fi2, there exists an infinite family

of edge-disjointuQi2-paths with distinct end-vertices onQi2, and we are done becauseE(H) ∩ E(Qi1) = ∅. �


2.3. Decompositions

A decompositionof a graphG is an equivalence relation onE(G) such that the subgraphinduced by the edges of any equivalence class is connected. The (edge-)induced subgraphsobtained in this way are called thefragmentsof the decomposition. Thus, a decompositionof a graphG may be considered as a family of edge-disjoint connected subgraphs ofGwhose union is the graphGminus its isolated vertices. Given a decomposition� of a graphG and a subgraphH, the�-shadow of His the subgraph ofG which is the union of allthe fragments of� that edge-intersectH. An induced subgraphA of a graphG is saidto be locally cycle-decomposable(resp.locally circuit-decomposable) in G if G\A has adecomposition into cycles andA-paths (resp. circuits,A-rays andA-paths) or, which isequivalent, ifG/A is cycle-decomposable (resp. decomposable into circuits andq

A-rays).

The decompositions relevant to our purposes are cycle decompositions (i.e. decom-positions whose fragments are cycles), circuit decompositions and decompositions intoedge-traceable graphs. Note that these three types of decompositions are closely related.Indeed it is a consequence of two classical results (Theorems2.24 and 2.25) that thecycle-decomposable graphs are exactly the graphs having a decomposition into finite edge-traceable graphs, and that the circuit-decomposable graphs are those that have a decompo-sition into locally finite edge-traceable graphs.In the finite case we haveVeblen’s Theorem that a finite graph has a cycle decomposition

if and only if it is eulerian. This immediately generalizes to the locally finite case:

Theorem 2.24.A locally finite graphhasa circuit decomposition if andonly if it is eulerian.

For arbitrary graphs we have:

Theorem 2.25(Nash-Williams[13] ). A graph has a decomposition into edge-traceablegraphs if and only if it is eulerian.

Recall that atransition systemon a graphG is a family� = (�x)x∈V (G) such that each�xis a partition into pairs of the set of edges incident withx (see[4, III.40]). Fromany transitionsystem�, one can construct the equivalence relation onE(G)which is the transitive closureof the relation given by all the pairs of all the�x ’s. Any class of that equivalence relationinduce then an edge-traceable graph, and the preceding theorem can therefore be stated inthe following stronger way:

Theorem 2.26(Sabidussi[15] ). Any transition system on an eulerian graph G induces adecomposition of G into edge-traceable graphs.

Graphs admitting a cycle decomposition have been characterized by Nash-Williams.This result—which we will refer to as Nash-Williams’s theorem—can be formulated asfollows:

Theorem 2.27(Nash-Williams[13] , Sabidussi[15] ). For any graph G, the following areequivalent:

(1) G has a cycle decomposition;


(2) G has no odd cut;(3) every finite subgraph of G is contained in a finite eulerian subgraph of G.

Since for any odd cut[A,A]G,A is always a disjoint union of regions, of which at least oneis odd, condition (2) can be replaced by(2′) G has no odd region.

Proposition 2.28. If a graphG has a decomposition� into circuits and rays such that eachray in � is �-dominated in G by its origin, and no odd cut of G separates two tails of adouble-ray belonging to�, then G is also cycle-decomposable.

Proof. Suppose the contrary.ThenbyNash-Williams’sTheorem,Ghasanodd cut[A,A]G.This implies that there exists a fragmentC ∈ � which contains an odd number of edgesof [A,A]G. However, it is easy to see thatC cannot be a cycle, that ifC is a double-ray,then [A,A]G must separate two tails ofC, and finally that ifC is a ray,[A,A]G mustseparate its origin from some of its tails. Hence in each of the three cases we contradict thehypothesis. �

Definition 2.29. Given two sets� and�′, both composed of edge-disjoint circuits andrays, we say that�′ has tails in� if each tail of each infinite fragment of�′ has a tailcontained in some fragment of�.

Lemma 2.30. Letu ∈ V (G) and� be a decomposition of G into circuits and u-rays. Thenevery finite set�′ of edge-disjoint circuits and u-rays ofG that has tails in� can be extendedto a decomposition�′ of G into circuits and u-rays such that� has tails in�′ and�′ hastails in�.

Proof. Without loss of generality we may suppose that the�-shadow of⋃C∈�′ C is G

itself. This implies, since�′ is finite and has tails in�, that� is finite, and thereforethatG is locally finite. Let�′′ be a maximal set of edge-disjointu-rays and double-raysof G\⋃C∈�′ C that has tails in�. PutH := ⋃

C∈�′∪�′′ C. It is easy to see that no tailof any infinite fragment of� is contained inG\H . Hence,� being finite andG locallyfinite,G\H is therefore an eulerian graph having no infinite component. ThusG\H has acycle decomposition and any such cycle decomposition together with�′ and�′′ form adecomposition�′ of G that has the desired properties.�

Lemma 2.31. Let G be a connected graph, x ∈ V (G), and� be a decomposition of G intocycles and exactly two x-rays. Then, for everyv ∈ V (G), there exists a decomposition of Ginto cycles and exactly twov-rays.

Proof. LetRandR′ be the twox-rays of�,P be anyxv-path ofG, andC1,C2, . . .,Cn thecycles of� that edge-intersectP. PutH := R ∪R′∪C1∪C2∪. . .∪Cn. SinceH is an infinitelocally finite connected eulerian graph, it contains av-rayQ. SinceH\Q is locally finiteand has exactly one vertex of odd degree (viz.v), it contains av-rayQ′. SinceH\(Q∪Q′)is eulerian and locally finite, by Theorem2.24, it has a circuit decomposition�′.


Now, observe thatH is the edge-disjoint union ofR1, R2 and the finite graphC1 ∪ C2 ∪. . . ∪ Cn. This implies that{Q,Q′} has tails in{R,R′}, and therefore thatH\(Q ∪Q′) isfinite. Thus,�′ is a cycle decomposition, and

�′′ := {Q,Q′} ∪ �′ ∪ (�\{C1, C2, . . . , Cn, R,R′})

is the desired decomposition ofG. �

3. Rays in cycle-decomposable graphs

In this section we look for rays that can be removed from (or added to) a cycle-decompos-able graph without affecting its cycle-decomposability.

Proposition 3.1. Let H be a cycle-decomposable graph,Ra raywhich is edge-disjoint fromH and�-dominated by its origin inH ∪ R. ThenH ∪ R is likewise cycle-decomposable.

Proof. As in the proof of Proposition2.28, suppose the contrary and consider an odd cut[A,A]H∪R. Since the cut[A ∩H,A ∩H ]H must be of even cardinality,E(R) must meet[A,A]H∪R an odd number of times, implying that a tail ofR is separated from its origin byan odd cut ofH ∪ R, contradicting the hypothesis.�

Hence in a decomposition into cycles and rays, the rays that are�-dominated by theirorigin can in somesensebe “melted” into the cycle-decomposable part. Conversely, onemayask if the removal of such rays from a cycle-decomposable graph will leave the remainderwith a cycle decomposition. The answer in general is no; nevertheless there exist rather widesufficient conditions, in particular Theorem 3.3, that allow one to find such “removable”rays. Note that such a ray must always be�-dominated by its origin because otherwise thecycle-decomposable graph would have an even cut that separates the origin of the ray fromsome of its tails. On the other hand, as is shown by the next proposition, we may be able to“remove” such rays in pairs.

Proposition 3.2. Let G be a cycle-decomposable connected graph, u ∈ V (G), and� be anon-�-dominated end of G. Then there exist two edge-disjoint u-raysS1 andS2 of � suchthatG\(S1 ∪ S2) is still cycle-decomposable.

Proof. Let R ∈ � be au-ray andH be the�-shadow ofR, where� is some cycle decom-position ofG. ThenH must be locally finite and every ray ofH is end-equivalent toR inH. Hence there is no finite set of edges ofH whose deletion fromH leaves more than oneinfinite component, which implies by Theorem2.1 thatH is a 2-way infinite trail, and henceis decomposable into two edge-disjoint 1-way infinite trailsP1 andP2, both starting atu.Being locally finite, each of the trailsPi is decomposable into cycles and exactly oneu-ray(saySi). HenceH\(S1 ∪ S2) is cycle-decomposable, and therefore so isG\(S1 ∪ S2). �

The next theorem shows that in most�-dominated ends, there exist rays that can beremoved, without affecting the cycle-decomposability of the graph.


R1

Uj

U

R

R′

Qj1

Qj2

Qj3

QI1 QI

3QI2 QI

4

Qj4

Fig. 2.

Theorem 3.3. Let G be a cycle-decomposable graph, u ∈ V (G), n ∈ � and� be an endof �-multiplicity > n that is �-dominated by u in G. Then G admits a decomposition intocycles and exactly n u-rays belonging to�.

The proof is based on the two following lemmas.

Lemma 3.4. Let G be a cycle-decomposable graph, u ∈ V (G), and{R,R′} be a2-tressethat consists of two u-rays R andR′, both�-dominated by u. LetK := R ∪R′, then thereexists a u-rayS ⊆ K such thatG\S is still cycle-decomposable.Moreover, given any countable family(Ri)i∈I of edge-disjoint rays ofG\K, that are

end-equivalent to R(and hence toR′) in G, S can be so chosen that there exists a u-rayS′ ⊆ K\S which is both�-dominated by u and end-equivalent to theRi ’s inG\S.

Proof. Let � be the end ofG that containsR, R′ and theRi ’s. Note that we may supposethatG is countable, because otherwise we could take any countable subgraph ofG in whichR,R′ and theRi ’s are still end-equivalent and�-dominated byu, and letH be the shadowof this subgraph with respect to some cycle decomposition ofG. It is easy to see that if theresult holds forH, it will also hold forG, becauseH andG\H are both cycle-decomposable,and end-equivalence and�-domination inH imply the same properties inG.We may furthermore assume that any ray of� edge-intersectsK or someRi , because

otherwise we could add additional rays to the familyRi , i ∈ I , to obtain a maximal one.SinceG is assumed to be countable, the extended family will also be countable.Now let U = {uj }j∈J be the set of all vertices that�-dominate� in G. Note thatJ is

countable and suppose for convenience that the index setsJ andI are disjoint.For eachi ∈ I , let (Qni )n∈� be an infinite family of vertex-disjointRiK-paths, and for

eachj ∈ J , let (Qnj )n∈� be a family of edge-disjointujK-paths whose endpoints inK areall distinct (see Fig.2). Since bothRandR′ are end-equivalent to eachRi and�-dominated


by eachuj inG, such(Qni )’s exist for anyi ∈ I ∪ J , and without loss of generality may beassumed to be edge-disjoint fromK (= R ∪R′).LetX = {xm}m∈� be an infinite set of vertices ofV (R) ∩ V (R′) such thatx0 = u and,

for anym ∈ �, xm is closer tou thanxm+1 on bothR andR′. Such a family is easilyconstructed by taking asx1 any vertex ofV (R) ∩ V (R′) − {u}, and asx2, any vertex ofV (R) ∩ V (R′) which is beyondx1 on bothRandR′, etc.Choose any function� : � → I ∪ J for which each element of the range is the image of

infinitely many elements of the domain.We will now recursively define two nested families(Wi)i∈� and(W ′

i )i∈� of uX-trails contained inR ∪R′ having the following properties: thereexists an increasing sequencem0 < m1 < m2 < · · · of non-negative integers such that, foreachi ∈ �, E(Wi+1) − E(Wi) andE(W ′

i+1) − E(W ′i ) are (not necessarily respectively)

the sets of edges of thexmi xmi+1-segments ofRandR′.SetW0 = W ′

0 = 〈u〉, where〈u〉 is the path whose only vertex isu, and supposeWk andW ′k have already been defined. Letxmk be the vertex�= u which is an endpoint of both

Wk andW ′k. Let nk be the smallestn for which the endpoint ofQn�(k) onR ∪R′ does not

belong toWk orW ′k. Denote this endpoint byyk and letmk+1 be the smallest subscriptm

for which yk belongs to thexmkxm-segment ofR or of R′, as the case may be. Denote byPk thexmkxmk+1-segment ofRand byP ′

k the corresponding segment ofR′.

If yk belongs toPk, then defineWk+1 := Wk∪P ′k and W ′

k+1 := W ′k∪Pk, and otherwise

defineWk+1 := Wk ∪ Pk and W ′k+1 := W ′

k ∪ P ′k. Finally putW := ⋃

k∈�Wk andW ′ := ⋃

k∈�W′k.

It is easy to see that bothW andW ′ are edge-disjoint 1-way infinite trails startingat u and thatW ∪ W ′ = R ∪R′. HenceW andW ′ are locally finite. Denote by〈u =x0, e0, x1, e1, . . . 〉 the 1-way infinite trailW, and letS := 〈u = xi0, ei0, xi1, ei1, . . . 〉 betheu-ray defined such that,i0 is the largest indexi for whichxi = u and such thatij+1 is thelargest indexi for whichxi = xij+1 (such indices exists becauseW is locally finite). Hence,for eachj ∈ �, the sub-trail〈xij+1, eij+1, xij+2, eij+2, . . . , xij+1〉 induce an eulerianfinite subgraph ofWedge-disjoint fromS. In fact,W\S is exactly the edge-disjoint unionof all such induced subgraphs. Since, by Theorem2.24, each of these finite subgraphs iscircuit-decomposable,W\S is also cycle-decomposable. Similarily, one can define anu-rayS′ ⊆ W ′ such thatW ′\S′ is cycle-decomposable. Now, let us prove thatSandS′ have thedesired properties. To do so we will show that:

(1) S′ andRi are end-equivalent inG\W for anyi ∈ I ;(2) uj �-dominatesS′ in G\W for anyuj ∈ U ;(3) G\W is cycle-decomposable.

Clearly, the result follows from Properties (1)–(3) becauseW\S is cycle-decomposable,G\W ⊆ G\S, andu ∈ U .By the construction ofW ′, for anyi ∈ I , there exist infinitely manyQni ’s for which one

endpoint belongs toW ′. This together with the fact thatW ′ is a locally finite 1-way infinitetrail implies thatS′ is end-equivalent toRi in G\W . We leave the details to the reader. Forsimilar reasons, one can also conclude that anyuj , j ∈ J , �-dominatesS′ in G\W .For the third statement, suppose the contrary. By Nash-Williams’s Theorem, there is an

odd cut[A,A]G\W with u ∈ V (A); and clearly we can supposeA to be connected. Since


u ∈ U , we infer from (2) thatA contains a tail ofS′, and therefore by (1) and (2),A containsU as well as a tail of eachRi . Moreover, it is easy to see that sinceG contains no odd cuts,Z := V (W) ∩ V (A) is infinite. Applying Proposition 2.11 withX := Z andG := A, weobtain either a vertexx ∈ V (A) and an infinite family ofxZ-paths ofA which pairwiseintersect inx only, or a rayS0 ⊆ A and infinitely many disjointS0Z-paths. In the firstcase, sinceW is a locally finite 1-way infinite trail that containsS, and sincex cannot beseparated from any infinite 1-way infinite sub-trail ofWby the removal of a finite numberof vertices ofW ∪A, we have thatx cannot be separated from a tail ofSby such a removal.Thusxmust dominateS in W ∪ A and hence inG, contradicting the fact thatA containsU, which contains all vertices that dominateS. In the second case, again becauseW is alocally finite 1-way infinite trail that containsS, similarly as for the first case, we have thatthe removal of a finite number of vertices ofW ∪ A cannot separate a tail ofS0 from atail of S. Thus,S0 is end-equivalent toS in A ∪ S (and therefore inG), whenceS0 ∈ �.SinceA contains a tail ofS′ and of eachRi and since[A,A]G\W is finite, it follows thatV (A) ∩ V (S′ ∪⋃i∈I Ri) is finite. Moreover, sinceS′ is contained in the one-way infinitetrail W ′, V (A) ∩ V (W ′ ∪ ⋃i∈I Ri) is also finite. SinceS0 ∈ � is contained inA, sometail of it is edge-disjoint from allRi ’s, fromW ′ and sinceS0 ⊆ A = (G\W) − A, alsoedge-disjoint fromW. Again this is a contradiction because it has been assumed that everyray of � edge-intersectsK (= R ∪R′ = W ∪W ′) or someRi . �

Lemma 3.5. Let � be any end of G. If a cycle-decomposable locally finite subgraph H ofG contains at least one ray of�, then H contains a2-tresse composed of rays of�.

Proof. Let R0 be any ray of� contained inH, andJ be the shadow ofR0 with respect tosome cycle decomposition ofH. This implies that no finite subset ofE(J ) separates anyray ofJ fromR0 and so, sinceJ is locally finite, no finite subset ofV (J ) separates any rayof J fromR0. Hence all rays ofJ are end-equivalent inJ. By Corollary2.9,J contains a rayR that meets every other ray ofJ infinitely often. SinceJ\R is locally finite and has exactlyone vertex of odd degree, it must contain a ray which together withR forms the desired pairof rays. �

Proof of Theorem 3.3.

Claim 1. Without loss of generality we may suppose that G is countable, and that everyvertex of infinite degree�-dominates�.

Let H1 be the union of somen + 1 edge-disjoint raysS1, . . . , Sn+1 of �, extendH1 toa new subgraphH2 by adding sufficiently many pairwise vertex-disjointH1-paths so thattheSi ’s all are end-equivalent inH2. It is easy to see that such paths exist because theSi ’sare end-equivalent inG. Note thatH2 is connected, is locally finite and has exactly oneend. Then extendH2 toH3 by adding countably manyuH2-paths which are pairwise edge-disjoint and edge-disjoint fromH2 such that theSi ’s are�-dominated byu in H3. Finally,letH4 be the shadow ofH3 with respect to some cycle decomposition ofG. ClearlyH4 iscycle-decomposable. Moreover, by construction, no vertex of infinite degree inH4 may beseparated fromu by a finite cutset ofH4. Sinceu �-dominates theSi ’s in H4, this implies


that any vertex of infinite degree ofH4 also�-dominates theSi ’s in H4. Then it is easy tosee thatH4 is countable, satisfies the conditions of the theorem (for the end�4 that containstheSi ’s) and that if the conclusion of the theorem holds forH4 with respect to�4, it willhold forGwith respect to�. So without loss of generality we may suppose thatG = H4.From now on, suppose by way of contradiction, thatG admits no decomposition into

cycles and exactlyn u-rays belonging to�. Suppose thatn is the smallest integer for whichthere exists a graph that satisfies the conditions of the theorem but not its conclusions. Sincesuch a graph is cycle-decomposable,n > 0.

Claim 2. Given anyn + 1 pairwise edge-disjoint rays of�, no two of these rays form a2-tresse.

By way of contradiction, suppose there exist edge-disjoint raysR1, R2, . . . , Rn+1 of �such thatR1 andR2 meet each other infinitely often. Since theRi ’s are�-dominated byuin G, we may suppose without loss of generality that they originate atu. Put

R := R1, R′ := R2 andI := {3,4, . . . , n+ 1},and letSandS′ be the twou-rays that are given in Lemma3.4. Thus the end�′ of G\Swhich contains the rayS′ is �-dominated byu in G\S and has a�-multiplicity > n − 1.Also by Lemma 3.4,G\S is cycle-decomposable; therefore, by the minimality ofn, G\Shas a decomposition into cycles and exactlyn− 1 u-rays belonging to�′, implying thatGhas the required decomposition.Denote byD the set of vertices that dominate� in G. ThenD is finite by Claim 2 and

Remark 2.4.

Claim 3. Let G′ be any(multi-)graph satisfying the conditions of the theorem and theconclusions of Claim1 and2. If {Ri}i∈I is a finite set of at least n edge-disjoint u-rays of�, then every odd region ofG\(⋃i∈I Ri) meets D.

By way of contradiction, suppose thatA is an odd region of

G1 := G\(⋃i∈I Ri),

which is disjoint fromD. Then there are two cases to consider.Case1: There exists a vertexv ∈ V (A) which is of infinite degree inA. Sincev is also

of infinite degree inG, v �-dominates� in G (Claim 1). This implies that there existsj ∈ Isuch thatv �-dominatesRj in Rj ∪A. However, sinceV (A)∩D = ∅,Rj is not dominatedinRj ∪A. Thus, by Proposition2.19(i)⇒ (iii ), it is easy to see thatRj ∪A contains at leasttwo edge-disjoint raysS1, S2 that meet each other infinitely often, and are end-equivalenttoRj in Rj ∪A (and hence inG). Hence,S1, S2 together with theRi ’s for i �= j contradictClaim 2.Case2:A is locally finite. LetA+ be the subgraph obtained fromA by adding the edges

in [A,A]G1, and let� = (�x)x∈V (G) be the transition system obtained from some cycledecomposition ofG. Note thatA+ also is locally finite. For eachx ∈ V (A), let �′

x be theset of all pairs{e, e′} ∈ �x such thate, e′ ∈ E(A+). Sinceu is of infinite degree inG (andhence inG1), u �∈ V (A). Thus, given anyx ∈ V (A), the degree ofx in

⋃i∈I Ri is even


(possibly zero). Moreover, the edges ofG1 incident withxwhich do not belong to any pairof �′

x are precisely those which are coupled in�x with an edge of someRi . Since�x isa pairing of all the edges ofG incident withx, it follows that the number of such edgeswhich belong toG1 (i.e. toA+) but do not belong to any pair of�′

x is even. Thus,�′x can

be extended to a (full) pairing�′′x of the set of edges ofA+ incident withx. Hence, the

pairings�′′x , x ∈ V (A), induce a decomposition�′′ ofA+ into edge-traceable graphs, finite

non-eulerian trails and 1-way infinite trails. Note that the last two categories of fragmentsof �′′ must have their initial and (in the case of finite non-eulerian trails) terminal edgesin [A,A]G1. Since[A,A]G1 is of odd cardinality, it follows that�

′′ must contain a 1-wayinfinite trail K. The transition system� being induced by a cycle decomposition, it is easyto see thatK must meet

⋃i∈I Ri (and hence someRi0) infinitely often.

Let x0 be any vertex ofV (K) ∩ V (Ri0) andH the union of the tail ofRi0 that begins atx0 and a 1-way infinite subtrail ofK that also starts atx0. Being a subgraph ofA+ ∪ Ri0,H is locally finite. By construction, it is one-ended, and hence cannot have any odd cuts.This implies by Nash-Williams’s Theorem thatH is cycle-decomposable. By Lemma3.5,H must contain two raysS1 andS2 which meet infinitely often and are end-equivalent toRi0 in G. ThusS1, S2 and{Ri}i∈I−{i0} again contradict Claim 2. This proves Claim 3.By Nash-Williams’s Theorem it is easy to see that a graph which is not cycle-decompos-

able must contain two vertex-disjoint odd regions. Hence, ifR1, . . . , Rn are edge-disjointu-rays of�, then by Claim 3 there exist two vertices inD which are not infinitely edge-connected inG\⋃n

i=1Ri . Thus to finish the proof, it suffices to constructn pairwise edge-disjointu-raysS1, . . . , Sn of � such that the vertices inD are still pairwise infinitely edge-connected inG\(⋃n

i=1 Si).LetR1, . . . , Rn+1 be pairwise edge-disjointu-rays of� and

L :=n+1⋃i=1

Ri.

We will now show that there exists a decomposition� of G\L into cycles and a finitenumber ofD-paths. It clearly suffices to find a decomposition ofG\L into cycles and anarbitrary number (possibly infinite) ofD-paths. Denote byG′ the multigraph obtained byadding toGa new vertexw and, for eachx ∈ D, a countably infinite set ofxw-edges. SinceD is finite, the vertices that dominate theRi ’s in G′ are exactly the vertices ofD (w doesnot dominate them). Also, inG′\L all the vertices ofD are still infinitely edge-connected.Moreover,G′\L does not have any odd cut because otherwiseG will contradict Claim 3.Hence by Nash-Williams’s Theorem,G′\L has a cycle decomposition and by removing theedges incident withw, we obtain the desired decomposition ofG\L.Now, choose a family(Qk)k∈� of vertex-disjointL-paths, and for eachv ∈ D, a family

(P vm)m∈� of vL-paths pairwise intersecting inv only such that

(a) theRi ’s are end-equivalent inL ∪ (⋃k∈�Qk);(b) eachPvm is edge-disjoint fromL ∪ (⋃k∈�Qk);(c) eachQk is internally vertex-disjoint fromL;(d) the�-shadows of theQk ’s are pairwise edge-disjoint and edge-disjoint from all the

D-paths of�.


R1

R2

R3

U

Fig. 3. Edges ofR: bold; edges belonging to someEv : whiskered.

We leave it to the reader to show that since bothD and the fragments of� are finite, onecan recursively construct such families.Denote byEv the set of all edges ofL that have a vertex in common with somePvm,

m ∈ �. Note thatEv is infinite for anyv ∈ D.Finally letRbe anyu-ray inL ∪ (⋃k∈�Qk) such that:

(1) Rcontains infinitely many edges ofEv for everyv ∈ D,(2) If we orient the edges of eachRi in the natural way (i.e. away fromu), then this will be

consistent with the natural orientation of the edges ofR.

(see Fig.3 for a simple example).We leave it to the reader to show that such a ray exists and that it follows fromProperty (2)

that the symmetric difference ofRandL containsn edge-disjointu-raysS1, . . . , Sn.To finish theproof, let us show that any two verticesa,bofDare infinitely edge-connected

inG\⋃ni=1 Si . Observe that since bothEa ∩E(R) andEb ∩E(R) are infinite, and sinceL

is locally finite,aandbmust be infinitely edge-connected in(G\L)∪R. DefineR′ to be the1-way infinite trail obtained fromRby replacing eachQk ⊆ R by the corresponding trailthat is induced by the edges of the�-shadow ofQk which do not belong toQk. Such anR′ exists because theQk ’s are assumed to be pairwise vertex-disjoint and their�-shadowspairwise edge-disjoint and edge-disjoint from theD-paths of�.Since bothRandR′ coincide onL and sinceEa andEb are contained inE(L), a andb

are also infinitely edge-connected in

G1 :=((G\L)\

⋃k∈�

Qk

)∪ R′.

MoreoverR′ is edge-disjoint from⋃k∈�Qk since by the choice ofR, eachQk is either

contained inRor edge-disjoint from it. ThusG1 ⊆ G\⋃ni=1 Si and we are done because

a andb are then infinitely edge-connected inG\⋃ni=1 Si . �

Observe that, given a cycle-decomposable graphGand an end of�-multiplicity n+1 thatis �-dominated by a vertexu, the maximal integerm for which there exists a decompositionof G into cycles and exactlym u-rays belonging to� is at mostn + 1; from this point of


view Theorem3.3 says thatm = n or n + 1. Our next proposition shows thatm = n + 1when� has exactly one dominating vertex.

Proposition 3.6. Let G be a cycle-decomposable graph, n ∈ �, and � be an end of�-multiplicity �n that has a unique dominating vertex u. Then for any vertexv that �-dominates�, G also admits a decomposition into cycles and exactly nv-rays belongingto �.

Proof. By Theorem3.3, we may assume that� has�-multiplicity exactlyn. If u �= v, notethat sinceu andv are infinitely edge connected and�-dominate�, it is easy to see thatwe only have to show the result for the case wherev = u. Let R1, . . . , Rn ben pairwiseedge-disjointu-rays of�, andH the�-shadow ofR1 ∪ . . . ∪ Rn, where� is some cycledecomposition ofG.We claim thatL := H\(R1 ∪ . . . ∪ Rn) is cycle-decomposable. Note that this claim

implies the result because any cycle decomposition ofL together with theRi ’s and thecycles of� that are edge-disjoint fromL form a decomposition ofG that has the desiredproperty. Thus, to finish the proof, suppose by way of contradiction thatL is not cycle-decomposable. By Nash-Williams’ theorem, there exists an odd regionA of L that doesnot containu. SinceH is cycle-decomposable,[A,H − A]H cannot be of odd cardinalityand hence must be infinite. This implies thatR1 ∪ . . . ∪ Rn (and hence someRi) meetsAinfinitely often. Then by Proposition 2.12,Amust contain a rayQ that is end equivalent toRi in G. HenceQ,R1, . . . , Rn aren + 1 pairwise edge-disjoint rays of�, implying that�has�-multiplicity > n, a contradiction. �

From another point of view, Theorem 3.3 also says that for eachk�m, G admits adecomposition into cycles and exactlyk u-rays belonging to�. The next proposition has asomewhat similar character.

Proposition 3.7. Let u ∈ V (G) and� be a decomposition of G into circuits and u-rayssuch that every ray contained in an infinite fragment of� is �-dominated by u in G. Let r, sbe respectively the number of rays and double-rays in�.Then, for any non-negative integern�r + 2s, there exists a decomposition of G into cycles and exactly n u-rays, each ray ofwhich is�-dominated by u in G.

Proof. By way of contradiction suppose thatn0 is the smallest value ofn for which thereexists a counterexample and thatG is such a counterexample.By Proposition2.28,G is cycle-decomposable, and thereforen0 > 0.Split each double-ray of� into two rays having the same origin, and let� be the set of

all these rays together with all the rays of�. Recall that|�|�n0. By Theorem 3.3, no endwhich is �-dominated byu in G contains more thann0 edge-disjoint rays. Henceu doesnot �-dominate any�-tresse, and it therefore follows from Proposition 2.22 that the partialorder(�,�u) has amaximal element, sayR. Thus, every ray of�−{R} is still �-dominatedby u in G\R. LetR′ be anyu-ray that shares a tail withR, put�′ := {R′}, and let�′ be adecomposition given by Lemma 2.30. Since the number of rays plus twice the number ofdouble-rays is the same in� and�′, it follows that�′ − {R′} is a decomposition ofG\R′


into circuits andu-rays that satisfies the condition of the proposition forn = n0−1. HenceG\R′ has a decomposition�′ into cycles and exactlyn0 − 1 u-rays�-dominated byu inG\R′. SinceR′ is �-dominated byu in G, the decomposition�′′ := �′ ∪ {R′} of G givesrise to a contradiction. �

Proposition3.7 together with Theorem 3.3 and the next result, although they deal withcycle-decomposable graphsonly,will be key results in our studyof the circuit-decomposabi-lity of arbitrary graphs.

Lemma 3.8. Let G be a cycle-decomposable connected graph andv ∈ V (G). If G has acircuit decomposition that contains at least one double-ray, then G has a decompositioninto cycles and exactly twov-rays.

Proof. First note that,G being connected, by Lemma2.31 it is enough to show that thereexists some vertexx ∈ V (G) for whichG admits a decomposition into cycles and exactlytwo x-rays. Let�0 be any cycle decomposition and�1 any circuit decomposition having atleast one double ray. There are two cases to consider.Case1: There is a regionB of G that contains a tail of some fragment of�1, and there

exists a vertexx ∈ V (G) that�-dominates all tails contained inB of infinite fragments of�1. (Note thatx ∈ V (B).) By Nash-Williams’s Theorem[B,B]G is an even cut. Thus thequotient graphG/B has a decomposition into circuits (essentially induced by�1) whichsatisfies the conditions of Proposition 3.7 forn = 2, i.e.,G/B has a decomposition�into cycles and exactly twox-rays, eachx-ray of� being�-dominated byx in G/B. Sincethe number of edges ofG/B incident withq

Bis∣∣[B,B]G

∣∣ (which is finite), and sincex�-dominates the two rays of�, B contains two edge-disjointx-rays that have tails in�.Therefore by Lemma 2.30,� may be so chosen that its twox-rays do not containq

B.

Define� as the set of all fragments of� that do not containqBtogether with all fragments

of �0 that are contained inB. Observe that every fragment of� (even those that belong to�) may be considered as a subgraph ofG, and thatG\(⋃C∈� C

)is cycle-decomposable

because it is eulerian and contains only a finite number of edges. Thus� can be extendedto a decomposition of all ofG into cycles and exactly twox-rays.Case2: G contains no region having the properties of Case 1. Denote byB the set of

all regions ofG that contain a tail of some infinite fragment of�1. For eachB0 ∈ B andy ∈ BdryG(B0) some fragment of�1 contains a rayR0 ⊆ B0 which is not�-dominated byy. Hence there exists a regionB ′ ∈ B such thatB ′ ⊆ B0 − y.SinceBdryG(B0) is finite, this implies that thereexists a regionB1 ∈ Bwhich is contained

in B0 and is disjoint from BdryG(B0). Repeating this argument, we obtain an�-stratifyingsequence(Bn)n∈� of G. Thus, by Proposition 2.17,G has a non-�-dominated end, andtherefore by Proposition 3.2, the required decomposition ofG exists. �

4. Having enough rays, a necessary condition

By Nash-Williams’s Theorem, a graph without odd cuts has a cycle decomposition andhencea circuit decomposition.Moreover, for anyodd cut[A,A]G of a circuit-decomposable


Fig. 4.

graphG, there must exist a transverse double-ray, i.e., a double-ray having a tail inA andanother one inA. This is because given any circuit decomposition� of G, the cyclesbelonging to� must meet[A,A]G an even number of times, and the same holds for thenon-transverse double-rays in�. This observation leads us to the following definition.

Definition 4.1. We say that a graphG hasenough raysif for each odd cut[A,A]G of G,bothA andA contain a ray.

It will be convenient to relativize this definition to various specific classes of rays (e.g.,non-dominated rays, removable rays) in the obvious way.In view of the symmetry of this definition and since for any odd cut[A,A]G,A has only

finitely many components that meet BdryG(A), each being a region ofG, not all of themeven, the definition can be equivalently stated as: a graphG hasenough raysif each oddregion contains a ray.In this language the remark at the beginning of this section becomes the following nec-

essary condition:

Proposition 4.2. A circuit-decomposable graph has enough rays.

Observe that this necessary condition implies that the graph is eulerian, because for anyvertex of odd degreex of G, [x,G− x]G is an odd cut and clearlyx does not contain rays.Unfortunately the condition stated in Proposition4.2 is not sufficient: see Fig. 4.The example of Fig. 4 contains two vertices of infinite degree; this is theminimal number

where the condition is not sufficient since, as will show Proposition 9.12, for graphs withat most one vertex of infinite degree the condition is both necessary and sufficient. This,incidentally, shows that the condition of having enough rays is strictly stronger than beingeulerian.

5. Dominated subgraphs

Before going further, we need to generalize the domination property which so far isdefined for rays only, to arbitrary subgraphs.


Definition 5.1. LetH be a subgraph ofG andx ∈ V (G).H is said to bedominatedby x inG if there exists an infinite set ofxH-paths pairwise intersecting inx only.

By Menger’s Theorem, this definition has the following equivalent form:

Lemma 5.2. A vertex x dominates a subgraph H in G if and only if for every finite set ofvertices S ofV (G)\{x} infinitely many vertices of H lie in the component ofG − S whichcontains x.

Proof. Left to the reader. �

As we will show later, from the point of view of circuit-decomposability, one of the mostinteresting classes of subgraphs are the non-dominated eulerian subgraphs ofG, because theremoval of suchasubgraph fromGdoesnot affect thedecomposability or indecomposabilityof G into circuits (Proposition8.1).The next proposition provides a useful tool for proving whether or not a subgraph is

dominated.

Lemma 5.3. LetHbe subgraph of a graphG.Then the following statements are equivalent:

(i) H is dominated in G;(ii) G contains a rayless tree T such thatV (T ∩H) is infinite;(iii) G contains a tree T and a vertexx ∈ V (T ) such that infinitely many different compo-

nents ofT − x meet H.

Proof. (i)⇒ (ii ). Immediate from the fact that the union of an infinite family ofxH-paths,pairwise intersecting inx only, is a rayless tree.(ii ) ⇒ (iii ). Let F be any rayless tree meetingH infinitely often. For each pairy, z ∈

V (H) ∩ V (F), letPyz be the uniqueyz-path inF, and letT ⊆ F be the union of allPyz’s.Note thatT is still an infinite rayless tree because it is clearly connected, included inF andcontains all vertices ofV (H) ∩ V (F). Therefore some vertexx has infinite degree inT.SinceT is a union of paths having their endpoints inH, every component ofT − x mustcontain a vertex ofH, and sincex is of infinite degree inT, there must be infinitely manysuch components.(iii ) ⇒ (i). It is easy to see that by taking onexH-path in each of the components of

T −x that meetH one obtains an infinite family ofxH-paths pairwise intersecting inxonly;i.e.H is dominated byx in G. �

Observe that every subgraphH of a graphG containing a vertexx of infinite degree inHis dominated byx in G. On the other hand, a connected locally finite subgraph is or is notdominated depending on whether or not it contains dominated rays.

Lemma 5.4. Let H be a subgraph of G that has a finite number of components. Then H isnon-dominated in G if and only if H is locally finite and does not contain any dominatedray of G.


Proof. Without loss of generality wemay suppose that bothH andGare connected becauseotherwise we can prove the result for each component ofH viewed as a subgraph of thecomponent ofG that contains it. SinceH has only finitely many components, this willimplies the result for the whole subgraphH.The necessity is a straightforward consequence of the definition. For the sufficiency,

suppose thatH is dominated inG and locally finite; let us show thatH must contain adominated ray. By Lemma5.3, letT be a rayless tree ofGmeetingH infinitely often, andlet J be a spanning tree ofH. Fix x0 ∈ V (H) and for eachx ∈ V (T ) ∩ V (H) let Px bethe x0x-path inJ. Then the union of these paths is infinite, connected and locally finite,and therefore must contain a rayR. Rmust meet infinitely manyPx ’s, and hence thereexist infinitely many disjointTR-paths inJ. Clearly these paths can be chosen so that eachincludes just one vertex ofT. Then the union of these paths andT is a rayless tree meetingR infinitely often, which by Lemma 5.3 gives the result.�

The following result relates the domination property of a subgraph to its cardinality.

Corollary 5.5. Let G be a graph. Then

(i) Every finite subgraph of G is non-dominated in G.(ii) Every non-dominated(not necessarily connected) subgraph of a connected graph G is

countable and locally finite.

Proof. (i) is immediate from the definition of dominance. For (ii), letH be any non-dominated subgraph ofG andT any spanning tree ofG. Let T ′ ⊆ T be the union ofall paths ofT having both endpoints inH. ClearlyT ′ is a tree and since for anyx ∈ V (T ′),each component ofT ′ − x intersectsH, Lemma5.3 implies thatT ′, and henceT ′ ∪ H ,is locally finite. SinceT ′ ∪ H is connected, it follows that it is also countable. ThusH islikewise locally finite and countable.�

6. Having enough non-dominated rays, a sufficient condition

The aim of this section is to show that “having enough non-dominated rays” is a suf-ficient condition for a graph to have a circuit decomposition or more precisely that thisproperty characterizes the graphs that have a decomposition into non-dominated circuits(Theorem 6.9). To do so we first have to restrict ourselves to the countable case; afterwards,we generalize to arbitrary cardinality using Theorem 6.8, which was proved in [11].The following definition provides a generalization of the notion of “having enough rays”.

Definition 6.1. A setR of ends ofG is said to bewell-spreadinG if each odd region ofGcontains a ray belonging toR.By extension we say thatR iswell-spreadin a subgraphH of G if each odd region ofH

contains a ray belonging toR.

As already mentioned after Definition4.1, a graphG has enough rays (resp. enoughnon-dominated rays) if and only if the set of all ends (resp. non-dominated ends) ofG iswell-spread.


The next lemma shows that the property of having enough non-dominated rays is resistantto the removal of non-dominated eulerian subgraphs.

Lemma 6.2. LetR be a well-spread set of non-dominated ends of G, and H a locally finiteeulerian subgraph of G that has at most finitely many connected components. If all rayscontained in H belong toR, thenR is also well-spread inG\H .

Proof. First note that by Lemma5.4,H is not dominated inG. By way of contradiction, letus suppose that there is an odd regionA of G\H that contains no ray ofR. There are twocases to consider.Case1: V (H) ∩ V (A) is infinite. Fixx0 ∈ V (A) and a spanning treeT of A and define

T ′ as the subtree which is the union of allx0H -paths ofT. SinceT ′ meetsV (H) infinitelyoften, Lemma 5.3 implies thatT ′ contains a rayR.Now let G be a new graph obtained fromG by the addition of a new vertexu joined

to each vertex ofR. It is easy to see thatH is dominated byu in G. Since the numberof components ofH is finite, one of them, sayK, is also dominated byu in G. Thus byLemma 5.4,K must contain a rayQ which is dominated inG. Since by hypothesisQ isnon-dominated inG, its dominating vertex inG must beu, implying, by the constructionof G, thatQ andRbelong to a same end ofG. This contradicts the fact thatQ, being a rayof H, belongs toR whereasRdoes not.Case2:V (H)∩V (A) is finite. By our notational convention,A is an induced subgraph of

G\H . Denote byA+ the vertex-induced subgraph ofG onV (A), and putA+ := G−A+.Observe thatH ∩ A+ must be finite becauseV (A) = V (A+) andV (H) ∩ V (A) is finite.Hence,H being locally finite,[A+, A+]G is likewise finite.Moreover,

∣∣E(H)∩[A+, A+]G∣∣

is even because otherwise[H ∩ A+, H ∩ A+]H would be an odd cut ofH, and therefore,G/A+ a finite graph having exactly one vertex of odd degree, which is not possible.To finish the proof, note that since∣∣∣[A+, A+]G

∣∣∣ =∣∣∣[A,A]G\H

∣∣∣+ ∣∣∣E(H) ∩ [A+, A+]G∣∣∣,

the cut[A+, A+]G ofG is odd, and sinceA (= A+\H ) contains no rays ofR andH ∩A+ isfinite,A+ is likewise free of rays ofR. This contradicts the hypothesis thatR is well-spreadin G. �

Lemma 6.3. LetR be a well-spread set of non-dominated ends of G. Then every edge ofG is contained in a circuit of G which is either finite or the union of two rays ofR.

Proof. We may supposeG to be connected sinceR is well-spread in each component ofG. Let e ∈ E(G). If e is contained in a cycle, then there is nothing to show. If this is notthe case, then{e} must be an odd cut, call it[A,A]G. SinceR is well-spread, letRA (resp.RA) be a ray ofR that is contained inA (resp.A). Since any two rays having a commontail are equivalent, we may suppose without loss of generality that the origin ofRA (resp.RA) is the only vertex ofA (resp.A) incident toe. ThenD := RA ∪ e ∪RA is a double-rayof G all of whose tails belong toR, andebelongs toD. �


Proposition 6.4. Let G be a countable graph andR a well-spread set of non-dominatedends of G. Then G has a circuit decomposition� such thatR is still well-spread in eachfragment of� (i.e., each double-ray of� is the union of two rays ofR).

Note that by Lemma5.4, the circuits of such a decomposition are all non-dominatedin G.

Proof. Let {e1, e2, . . .} be an enumeration ofE(G). We will construct a circuit decom-position(Ci)i∈I inductively in the following way: suppose that for every positive integerk < j, Ck has already been constructed and thatCk is either finite or the union of two raysof R. Let ij be the smallest subscript such thateij �∈ ⋃

k<j E(Ck). If no such subscriptexists then(Ci)i<j is the desired circuit decomposition. Otherwise, by Lemma 6.2,R iswell-spread inG\(⋃k<j Ck). By Lemma 6.3, letCj ⊆ G\(⋃k<j Ck) be any circuit whichis either finite or the union of two rays ofR and contains the edgeeij . Clearly(Ck)k<r is,for somer��, a circuit decomposition ofG that has the desired property.�

The generalization of this last result to the uncountable case is an immediate consequenceof the following key result.

Theorem 6.5. LetR be a well-spread set of non-dominated ends of G. Then G is decom-posable into countable fragments in each of whichR is still well-spread.

For the proof of this theorem we have to recall some definitions and a result from Part Iof this paper[11].

Definition 6.6. An �-decompositionof a graph is a decomposition whose fragments are allof cardinality less than or equal to�.

Definition 6.7. An �-decomposition� of a graphG is said to bebond-faithfulif

(i) any bond ofG of cardinality�� is contained in some fragment of�;(ii) any bond of cardinality< � of a fragment of� is also a bond ofG.

Since any cut is the edge-disjoint union of bonds, condition (i) implies that condition (ii)can be replaced in an equivalent manner by:

(ii ′) any cut of cardinality< � of a fragment of� is also a cut ofG.

Theorem 6.8(Laviolette[11] ). Let(Hi)i∈I beapairwiseedge-disjoint family of countableconnected subgraphs of G. Then G has a bond-faithful�-decomposition� such that eachHi and each non-isolated vertex of degree�� in G is contained in one and only onefragment of�.

Assuming the Generalized Continuum Hypothesis, this last result can be generalized to�-decompositions with� > �. See[11].


Proof of Theorem 6.5. Let U be a maximal set of pairwise vertex-disjoint rays ofR. Weclaim that for every odd regionA of G there exists a ray inU that has a tail inA. Supposethat an odd regionA of G does not contain the tail of any ray inU . Since[A,A]G is finite,V (R∩A) is finite for everyR ∈ U . Moreover, since the rays inU are pairwise edge-disjoint,at most

∣∣[A,A]G∣∣ rays inU meetA, and soV (A ∩⋃R∈U R) is finite. Hence any ray inA

has a tail which is disjoint from all rays inU . This contradicts the maximality ofU , sinceA contains a ray belonging toR, and any tail of a such a ray is likewise inR.Let G be the graph obtained fromGby the addition of a new vertexu joined to the origin

of each ray inU , and for anyR ∈ U , consider the rayR := eR ∪R,

eR being the edge joiningu to the origin ofR. Let � be a bond-faithful�-decomposition ofG such that for anyR ∈ U , R is contained in a fragment of�. Such a decomposition existsby Theorem6.8,(R)R∈U being a family of pairwise edge-disjoint graphs of cardinality atmost�.

Claim. Wemay suppose that no fragment of� has u as a cut-vertex. Otherwise decomposeeach fragmentH ∈ � into (Hi)i∈IH ,where eachHi is a component ofH −u together withu and all the edges ofH joining u to a vertex of that component. Note that if u is not acut-vertex ofH , then|IH | = 1.Consider the�-decomposition� of Gwhose fragments areall theHi ’s,whereH runs through all fragments of�. Since every rayR (R ∈ U) containsexactly one edge incident with u, it is easy to see that each suchR is still contained in oneand only one fragment of�.

In any graph, no two edges of a bond are separated by a cut-vertex. It follows that foreachH ∈ � the set of all bonds ofH is exactly the set of all bonds of all theHi ’s (i ∈ IH ).Thus,� is a bond-faithful decomposition ofG, and therefore can play the role of�, provingthe Claim.Now, let

� := (H − u)H∈�,

whereH − u meansH if u �∈ V (H ). Clearly,� is an�-decomposition ofG since nofragment of� containsu as a cut-vertex. Let us prove that� is the desired decompositionof G (i.e.R is well-spread in each fragment of�).LetH be any fragment of�, H the corresponding fragment in� andC = [A,H −A]H

any odd cut ofH. To finish the proof, let us show thatA contains a tail of a ray ofU . By wayof contradiction suppose that no ray ofU has a tail inA. This implies that at most a finitenumber of the rays ofU which are contained inH (in fact at most|C|) have their origin inA.HenceC := [A, H −A]H is a finite cut ofH becauseC consists ofCand all edges joiningu to a vertex inAwhich is the origin of some ray inU . By the bond-faithfulness of�, C isalso a cut ofG. ThereforeC = [B, G− B]G for someB ⊆ G and the notation may be sochosen thatA ⊆ B andH − A ⊆ G− B. Then it is easy to see thatA = B ∩H , whence

[A,H − A]H = C = [B − u,G− B]G.


HenceC is also an odd cut ofG, which implies that there exist two raysQ,Q′ ∈ Uwhich respectively have a tail inB andG − B. Sinceu is the origin of bothQ andQ′,C (= [B, G − B]G) contains an odd number of edges (hence at least one) of eitherQ orQ′, according asu is a vertex ofG − B or B. By the definition of�, this particular ray istherefore contained inH becauseC is a cut ofH andH ∈ �. SinceA = B ∩H , the casewhereu ∈ V (G − B) andQ ⊆ H gives rise to a contradiction becauseQ (= Q − u),having a tail inB, will have a tail inA. For the case whereu ∈ V (B) andQ′ ⊆ H , thevertexu, being the origin ofQ′ is a vertex ofH , a contradiction tou �∈ V (A) andH −A ⊆G− B. �

In viewofTheorem6.5,Proposition6.4 isalso truewithout theassumptionof countability.In particular, ifR is the set of all the non-dominated ends, we have the following.

Theorem 6.9. A graph has a decomposition into non-dominated circuits if and only if ithas enough non-dominated rays.

Proof. The necessity is evident and the sufficiency is a consequence of Proposition6.4 andTheorem 6.5. �

Hence the property of having enough non-dominated rays is a sufficient condition for agraph to admit a circuit decomposition. Thus for any class of graphs in which all rays arenon-dominated, the property of having enough rays is a necessary and sufficient condition.One such class is formed by the block-locally-finite graphs (i.e. the graphs all of whoseblocks are locally finite). These graphs are interesting because they are a generalization oftrees and of locally finite graphs. Indeed, they have the following characterization.

Lemma 6.10. A graph is block-locally-finite if and only if it contains no dominated rayand no pair of infinitely edge-connected vertices.

Proof. The necessity is obvious and the sufficiency a consequence of Proposition 2.11.�

Another family of graphs for which having enough rays is a necessary and sufficientcondition for circuit-decomposability, are those that have at most one vertex of infinitedegree, see Proposition9.12.

7. Peripheral regions, odd-type vertices and circuit decomposition

7.1. Peripheral regions

Intuitively speaking, a ray that goes through infinitely many “successive” odd cuts, as inFig. 5 is always non-dominated. Hence in a graph which is not circuit-decomposable, andtherefore, by Theorem 6.9, which does not have enough non-dominated rays, there mustbe odd regions that are “poor” in odd cuts. To make this idea precise, let us first define thefollowing type of region:


Fig. 5.

B

A A

Fig. 6.

Definition 7.1. An odd region of a graphG is peripheralif it contains no odd cut ofG.

The word peripheral has been chosen because such regions can be visualized as lying onthe “periphery” of the drawing of the graph. Note that a peripheral regionAmay contain asubgraphB such that[B,B]G is an odd cut, but if this is so, then[A,A]G ∩ [B,B]G �= ∅(see Fig.6). Moreover, ifB is a region (i.e. connected), then it is automatically peripheral.

Proposition 7.2. Every odd regionofGcontains aperipheral regionofGor an�-stratifyingsequence of odd regions of G.

Proof. Suppose thatG has an odd regionA0 which contains no peripheral region. SinceA0 itself is not peripheral, there exists an odd regionA1 ⊆ A0 of G such that[A1, A1]G ⊆E(A0) (and hence such thatA1 ⊆ A0 − Bdry(A0)). Repeating this argument ad infinitum,one can construct the desired�-stratifying sequence(Ai)i∈�. �

Definition 7.3. A region of a graph isobstructiveif all of its rays are dominated. (Inparticular a region which contains no rays is obstructive.)

The next three results will show that obstructive peripheral regions are the only parts ofthe graph where one may expect to encounter serious difficulties in connection with circuitdecompositions. Recall that if a graph does not have enough non-dominated rays, then bydefinition it has an odd region all of whose rays are dominated.


Proposition 7.4. Every obstructive odd region contains a peripheral region.

Proof. Propositions7.2, 2.17 and Remark 2.16�

Proposition 7.4 and Theorem 6.9 together say that a graph having no peripheral regionis circuit-decomposable. The next theorem points out, in a more specific way, the link thatexists between the structure of the peripheral regions and the existence or non-existence ofa circuit decomposition of the graph.

Theorem 7.5. Let G be a graph. Then the following statements are equivalent.

(i) G is circuit-decomposable;(ii) every peripheral region of G is locally circuit-decomposable;(iii) every obstructive peripheral region is locally circuit-decomposable in G;(iv) among all maximal families(Ai)i∈I of vertex-disjoint obstructive peripheral regions,

there is one which consists of locally circuit-decomposable regions.

Proof. (i)⇒ (ii )⇒ (iii )⇒ (iv) are evident.(iv)⇒ (i). First observe thatwithout lossof generalitywemaysupposeG to beconnected

and eulerian because for any vertexxof odd degree,{x} is a rayless peripheral region that isnot locally circuit-decomposable inG, and neither is any peripheral region containingx. Let(Ai)i∈I be a maximal family of vertex-disjoint obstructive peripheral regions and supposethat eachAi is locally circuit-decomposable. For eachi ∈ I , let Li be a finite connectedsubgraph ofAi that contains all the vertices in BdryG(Ai), and�i be a decomposition ofAi ∪ [Ai,Ai]G into circuits,Ai-rays andAi-paths. LetKi be the subgraph ofAi that is theunion of all the fragments of�i which are edge-disjoint fromLi ∪ [Ai,Ai]G. Note thatKiis circuit-decomposable, and thatG\Ki is still eulerian. Since theAi ’s are vertex-disjoint,so are theKi ’s; henceH := G\(⋃i∈I Ki) is eulerian. We claim thatH has enough non-dominated rays. Note that if this claim is true then we are done, since by Theorem6.9,His circuit-decomposable, and hence, theKi ’s being pairwise disjoint,(Ki)i∈I ∪ {H } is adecomposition ofG into circuit-decomposable graphs.To prove the claim, let us suppose by way of contradiction thatH does not have enough

non-dominated rays. By Proposition 7.4, there is a peripheral regionB of H that containsno non-dominated rays. Now put

Bi := Ai\Ki (= Ai ∩H), i ∈ I.SinceBi ∪ [Ai,Ai]G is the�i-shadow ofLi ∪ [Ai,Ai]G for any i ∈ I , we have that[Bi, Bi]H = [Ai,Ai]G and thatBi is infinite, locally finite and connected for anyi ∈ I .There are now two cases to consider.Case1: B ∩ (⋃i∈I Bi) is infinite. In that case, there must exist ani0 ∈ I such that

Bi0 ∩ B is infinite because otherwise, eachBi being infinite and connected, an infinitenumber of themwould have an edge in the finite cut[B,B]H , contradicting the fact that theBi ’s are pairwise disjoint. SinceBi0 is connected and[B,B]H is finite,Bi0 ∩ B contains afinite number of components. One of these components is therefore infinite. This particularcomponent contains a ray (sayR) because it is an infinite connected subgraph of the locally


finite graphBi0. Being contained inBi0, which is a locally finite region ofH,R is thereforenot dominated inH, a contradiction.Case2: B ∩ (⋃i∈I Bi) is finite (possibly empty). LetD := B ∩ (⋃i∈I Bi). Since

theBi ’s are pairwise disjoint locally finite regions of the eulerian graphH, every vertex

of D is of even degree inH. This implies that∣∣∣[D,D]H

∣∣∣ is even because it is equal to∑x∈V (D) degH (x) − 2

∣∣∣E(D)∣∣∣. Hence, by Lemma2.2, [B − D,B −D]H is an odd cut.

Being contained inB, B − D must by Proposition 7.4 contain an obstructive peripheralregionC of H. However,C is disjoint from eachAi, i ∈ I , implying that it is an inducedsubgraph ofG, and hence an obstructive peripheral region ofG, a contradiction to themaximality of the family(Ai)i∈I . �

7.2. The parity type of vertices

Observe that in a finite and also in an infinite locally finite graphG, a peripheral regionA always contains an odd number of vertices whose degree inG is odd, and these verticesmust belong to BdryG(A). In general, this is not true because of the vertices of infinitedegree, which in a sensemay be considered of both odd and even degree. This double statusis clearly the basic reason why circuit decompositions become so difficult to study once wego beyond the locally finite case. The following is a generalization to vertices of infinitedegree of the parity property that exists for vertices of finite degree.

Definition 7.6. A vertexx of a graphG is said to be ofeven typein G if every odd regionof G that containsx also contains an even region which still containsx. A vertex which isnot of even type is said to be ofodd type.

Note that any two infinitely edge-connected vertices are always of the same type, hencewe can define aneven-type class(resp.odd-type class) as an�-class whose members areof even type (resp. odd type).It is easy to see that verticesof evenor odddegreeare respectively of even typeor odd type.

Moreover, as shownby the results of the rest of Section7, there is a close connectionbetweenodd-type vertices and peripheral regions (the “problematic” parts of the graph from thepoint of view of circuit-decomposability) and between even-type vertices and locally cycle-decomposable regions (the “nice” parts). Moreover, since by Nash-Williams’s Theoremlocally cycle-decomposable regions cannot contain odd cuts, they are in some sense theeven counterparts of the peripheral regions.

Proposition 7.7. Anodd-typevertexalwaysbelongs to someperipheral regionof thegraph.

Proof. Without loss of generality suppose thatG is connected and by way of contradiction,let x be an odd-type vertex ofG that does not belong to any peripheral region ofG. By thedefinition of odd type,G has an odd regionA that containsx and such that no even regioncontainingx is contained inA. Now observe that if there exists an odd cut[C,C]G ⊆ E(A)such thatA ∩ C is connected and contains all the vertices of BdryG(A), thenA ∩ C is an


even region because in such case

[A ∩ C,A ∩ C]G = [A,A]G ∪ [C,C]G.Thus to reach a contradiction, let us construct such a cut. Denote byX the�-class ofx inG,and choose a finite treeT ⊆ A that containsx and every vertex of BdryG(A). LetB ⊆ A

be a region that containsx but that is vertex-disjoint fromT − X; by Lemma2.3, such aregion exists becauseT is finite and no vertex ofT −X is infinitely edge-connected tox inG. Note that by assumption,Bmust be an odd region but not a peripheral one. Thus thereexists an odd cut[C,C]G ⊆ E(B) with x ∈ V (C). Since a cut is an edge-disjoint union ofbonds, we may suppose without loss of generality that[C,C]G is a bond and therefore thatC is connected.T is contained inC because the only edges ofT that belong toB are thosethat have both endpoints inX, and therefore must belong toC. Hence sinceT is connectedand contained inA, and since

{x} ∪ BdryG(A) ⊆ V (T ),A ∩ C is likewise connected and containsx and the vertices of BdryG(A). This completesthe proof. �

Corollary 7.8. The vertices of a cycle-decomposable graph are all of even-type.

The next result shows that in appropriate circumstances parity type is preserved underthe operation of taking quotients.

Lemma 7.9. Let A be any induced subgraph of G such that[A,A]G is finite, and letx ∈ V (A). Then x is of even type in G if and only if it is of even type inG/A.

Proof. (⇒:) By way of contradiction suppose thatx is of even-type inGand an odd regionB ofG/A containsx but contains no even region ofG/A which containsx. Without loss ofgenerality we may suppose thatB ⊆ A (i.e.,q

A�∈ V (B)) because otherwise, degG/A(qA)

being finite, the component ofB − qAthat contains the vertexx will be a region ofG/A

contained inB, and hence an odd region that contains no even region that containsx. ThusB is also an odd region ofG that containsx. Now sincex is of even type inG, there existsan even regionB ′ of G such thatx ∈ B ′ ⊆ B, which gives rise to a contradiction becauseB ′ is also an even region ofG/A.(⇐:) Again by way of contradiction suppose thatx is of even-type inG/A andB is an

odd region ofGwhich containsx but whose even subregions do not containx. This impliesthat the componentK of B ∩ A that containsx is an odd region ofG, and hence clearly anodd region ofG/A. Nowwe obtain our contradiction by taking any even region ofG/A thatcontainsx and is contained inK, any such even region ofG/A being also an even region ofG. �

We now establish the property—already mentioned in the introduction—that the paritytype of a vertex does not change if we remove a finite eulerian subgraph. In fact the removalof an arbitrary eulerian subgraphH of Gmay only change the parity type of the vertices


that�-dominateH inG, where the concept of�-domination of a subgraph (analogous to theconcept of domination introduced in Section5) is defined as follows: a vertexx is said to�-dominate a subgraph H of Gif there is an infinite family of edge-disjointxH-paths havingdifferent end-vertices inH, or equivalently, if every region ofG that containsx, containsinfinitely many vertices ofH.

Proposition 7.10. Let H be any eulerian subgraph of G and x any vertex that does not�-dominate H in G. Then x is of even type in G if and only if it is of even type inG\H .

Proof.Case1:H is finite.(⇒:) LetA be any odd region ofG\H that containsx, and let us show thatA contains an

even region that containsx. By Lemma2.3, choose a regionB ⊆ A such thatV (B)∩V (H)is contained in the�-class ofx. If B is even we are done. So suppose thatB is odd (inG\H ).LetB+ be the subgraph ofG induced byV (B) (i.e.,B+\H = B). SinceH is eulerian andfinite, B+ is an odd region ofG. Consequently, there exists an even regionC of G thatis contained inB+ and containsx. Clearly,[C\H,C\H ]G\H is an even cut. Moreover,Cbeing connected, so also isC\H because otherwise there exists an edgee ∈ E(C)∩E(H)whose two incident vertices (x1, x2, say) belong to different components ofC\H . But thisis a contradiction because, being in the same�-class ofG, x1 andx2 cannot be separatedby the removal of the edges constituting the finite setE(H) ∪ [C,C]G. ThusC\H is thedesired even region ofG\H .(⇐:) LetA be any odd region ofG, andB the component ofA\H that containsx. Since

[A\H,A\H ]G\H is finite,B is therefore a region ofG\H . LetD be an even region ofG\Hthat is contained inB and containsx (if B is already an even region, putD := B). Then thesubgraph ofG induced byV (D) is an even region ofG contained inA that containsx.Case2:H is infinite. Sincexdoes not�-dominateH, there is a regionAofG that contains

x and such thatA ∩H is finite. It follows from that finiteness that

degH/A(qA) =∑

x∈V (A∩H)degH (x) − 2

∣∣∣E(A ∩H)∣∣∣,

and hence thatH/A is a finite eulerian subgraph ofG/A (possibly a single vertex). More-over,[A\H,A\H ]G\H is a finite cut. Hence we obtain thatx is of the same parity type inG, inG/A, in (G/A)\(H/A) ( = (G\H)/(A\H)), and inG\H , by applying Lemma7.9to G andA, the present proposition (Case 1) toG/A andH/A, and Lemma 7.9 toG\HandA\H . �

7.3. Vertices of even type and locally cycle-decomposable regions

Lemma 7.11. Let X be an�-class of G contained in some peripheral region A of G, andletB ⊆ A be a region that contains X but no vertex ofBdryG(A)−X. Then the followingstatements are equivalent.

(1) X is of even type;


AA

B

C

X

Fig. 7.

(2) B is an even region of G;(3) B is locally cycle-decomposable in G.

Note that since BdryG(A) is finite andX is an�-class ofG, Lemma2.3 implies that therealways exists a regionB ⊆ A such thatX ⊆ V (B) andV (B) ∩ BdryG(A) ⊆ X.

Proof. (3) ⇒ (1). Observe that, sinceG/B is cycle-decomposable, by Nash-Williams’sTheoremXmust be of even type inG/B and hence, by Lemma 7.9, of even type inG.(2) ⇒ (3). By way of contradiction, suppose thatB is an even region that is not locally

cycle-decomposable. ByNash-Williams’s Theorem, there is an odd cut[C, (G/B)−C]G/Bof G/B such thatX ⊆ V (C). Note that we may suppose without loss of generality thatqB

∈ V (C), because otherwise consider the cut induced byC ∪ {qB} instead ofC: by

Lemma 2.2, the new cut will then also be odd sinceqBis a vertex of even degree inG/B.

SinceqB

∈ V (C), it follows thatF := (G/B)−C is a subgraph ofG. Moreover[F,F ]G =[C,G/B − C]G/B and so[F,F ]G is an odd cut ofG that is contained inB ∪ [B,B]G,and hence inA ∪ [A,A]G. Moreover, sinceV (B) ∩ BdryG(A) ⊆ X andV (F) ∩ X = ∅,no edge of[F,F ]G ∩ [B,B]G belongs to[A,A]G. Thus[F,F ]G is totally contained inA,contradicting the fact thatA is peripheral inG.(1) ⇒ (2). By way of contradiction suppose thatB is an odd region ofG. SinceX

is assumed to be of even type, there exists an even regionC of G that containsX and iscontained inB. PutD := B − C. As one can easily see in Fig. 7, this implies that∣∣∣[D,D]G

∣∣∣ =∣∣∣[B,B]G

∣∣∣− ∣∣∣[C,B]G∣∣∣+ ∣∣∣[C,C]G

∣∣∣− ∣∣∣[C,B]G∣∣∣.

Since∣∣∣[B,B]G

∣∣∣and∣∣∣[C,C]G∣∣∣are of different parity,[D,D]Gmust be anodd cut. However,

sinceC containsX, andB contains no element of BdryG(A)−X, it follows thatA containsthe odd cut[D,D]G, a contradiction sinceA is peripheral. �

The preceding result establishes a certain link between even-type vertices and locallycycle-decomposable regions of the graph. This link comes out in an even stronger way inthe following proposition which is a generalization to the infinite case of a basic propertyof the vertices of even degree in finite graphs.


Proposition 7.12. Let B be a region of a graph G. Then

B is locally cycle-decomposable⇒ each vertex ofB is of even type inG.

Moreover, if B is contained in some peripheral region of G, the converse is also true.

Proof. If B is locally cycle-decomposable then, by Nash-Williams’s Theorem and sinceG/B is cycle-decomposable, each vertex ofBmust be of even type inG/B, and hence byLemma7.9, of even type inG. Now, suppose thatB is composed of even type vertices ofGonly, that it is contained in some peripheral regionA, and let us show that it is locally cycle-decomposable. By way of contradiction, suppose thatG/B is not cycle-decomposable, andby Nash-Williams’s Theorem, letCbe an odd region ofG/B such thatqB �∈ V (C). ThenCis also an odd region ofG contained inB, and hence inA. SinceA is peripheral,[C,C]G �⊆E(A), which implies thatC ∩BdryG(A) is non-empty. Since moreover it is finite, withoutloss of generality, wemay supposeChas been chosen such thatC∩BdryG(A) has smallestpossible cardinality, and letx ∈ C ∩ BdryG(A). Sincex is of even type inG and lies inC,it is contained in some even regionDx ⊆ C of G. By Lemma 2.2,[C −Dx,C −Dx]G isan odd cut, and sinceC − Dx is a union of disjoint regions, there therefore exists an oddregionCx of G that is a component ofC −Dx . Clearly,Cx is an odd region ofG/B suchthatqB �∈ V (C), a contradiction to the minimality assumption because

Cx ∩ BdryG(A) ⊆ C −Dx ∩ BdryG(A)�C ∩ BdryG(A). �

7.4. Vertices of odd type and peripheral regions

The next proposition plays the same role for odd-type vertices as Proposition7.12 doesfor the vertices of even type. In other words, it shows that some properties that are satisfiedby the vertices of odd degree in a finite graph are also satisfied by the vertices of odd typein an arbitrary graph, and also points out a link that exists between odd-type vertices andperipheral regions.

Proposition 7.13. Let G be a graph. Then

(i) every peripheral region A contains an odd number of odd-type classes and each suchclass meetsBdryG(A);

(ii) for every odd-type class O and every region A that contains O there exists a peripheralregionB ⊆ A such that O is the only odd-type class of G to be contained in B.

The proof of this proposition is based on the following lemma.

Lemma 7.14. Let A be a peripheral region of G and(Pi)i∈I be a maximal set of edge-disjoint paths in A such that the two endpoints of eachPi belong toBdryG(A) but arein different�-classes of G. ThenH := (

⋃i∈I Pi) ∪ [A,A]G is finite, A\H is cycle-

decomposable, and for each�-class X contained in A we have:

X is of odd type inG⇐⇒X contains an odd number of vertices

whose degree inH is odd.


Note that in the case where BdryG(A) is contained in a single�-class ofG (in particularwhen[A,A]G is a bridge), Lemma7.14 asserts thatA is cycle-decomposable. (See Fig. 9at the end of the paper for some examples.)

Proof. Clearly I is finite because so is BdryG(A) and because between two different�-classes ofG there is only a finite number of edge-disjoint paths. HenceH is finite.Let K be the subgraph ofG induced byV (A) ∪ BdryG(A). SinceA is peripheral in

G,G/K has no odd cut and hence, by Nash-Williams’s Theorem, is cycle-decomposable.Moreover, sinceI is finite andPi/K a cycle or an edge-disjoint union of several cycles,for any i ∈ I , (G −⋃

i∈I Pi)/K is also cycle-decomposable. AssumeA\H is not cycle-decomposable. Since(A\H)/K = (G−⋃i∈I Pi)/K, there exist two vertices of BdryG(A)that belong to the same component ofA\H but which are separated in that component byan odd (and hence finite) cut, contradicting themaximality of the family(Pi)i∈I . ThusA\His cycle-decomposable.Now letX be any�-class ofG that is contained inA, and letB ⊆ A be any region ofG

that containsXbut no vertex of BdryG(A)−X.As noted after the statement of Lemma 7.11,such aB exists. Moreover, sinceA\H contains no odd cut,B is an odd region if and onlyif X contains an odd number of vertices of odd degree inH. Thus by Lemma 7.11 we aredone. �

Proof of Proposition 7.13. (i) Take any finite subgraphH as defined in Lemma 7.14, andputG1 := (G\A) ∪ H . SinceH is finite and[A ∩ H,G1 − (A ∩ H)]G1 is an odd cut ofG1,A∩H contains an odd number of vertices whose degree inG1 is odd. Since vertices ofA∩H have the same degree inH as inG1, it follows that

∣∣V (A)∩ Vodd(H)∣∣ is odd whereVodd(H) is the set of vertices whose degree inH is odd. Therefore there are an odd numberof �-classesX ⊆ V (A) such that

∣∣X ∩ Vodd(H)∣∣ is odd. By Lemma 7.14, these�-classes

are precisely the odd-type classes contained inA. Moreover, ifX is one of these�-classes,then∅ �= X∩Vodd(H) ⊆ BdryG(A) because vertices inV (A∩H)−BdryG(A) have evendegree inH. This proves part (i) of Proposition 7.13.(ii) By Proposition 7.7, there is a peripheral regionC of G that containsO. Since, by

Lemma 2.2,[C∩ A,C ∩ A]G is finite, there exists a regionD ofG such thatO ⊆ V (D) ⊆V (C ∩ A). By Lemma 2.3 (withA,Y, x replaced byD, BdryG(C) − O, and an arbitraryelement ofO, respectively), there exists a regionB of G such thatB ⊆ D,O ⊆ V (B) and

B ∩ BdryG(C) ⊆ O. (1)

SinceB is contained inD, it is contained in the peripheral regionC. Hence, it follows fromLemma7.11 (withX,A, B replaced byO,C, B, respectively) thatB is an odd and thereforeperipheral region ofG. Moreover, it follows from (i) (withA replaced byC), from Eq. (1)and from the fact that an�-class is either disjoint fromV (B) or contained in it, thatBcontains no odd-type class other thanO. Thus we are done.�

Proposition 7.13 says in particular that a graph which has no odd-type vertex does nothave any peripheral region. Thus by Theorem 7.5 we have the following corollary of Propo-sition 7.13.


Corollary 7.15. A graph all of whose vertices are of even type is circuit-decomposable.

For vertex transitive graphs, Corollary7.15 has the following interesting consequence.

Corollary 7.16. A vertex transitive graph is circuit-decomposable if and only if it is eule-rian.

Proof. Let G be an eulerian vertex transitive graph. By Corollary7.15, we only have toshow that every vertex ofG is of even type. Assume the contrary. Then sinceG is vertextransitive, every vertex ofG is of odd type. LetAbe a peripheral region that contains exactlyone odd-type class ofG. Denote this class byO and observe thatO = V (A). This impliesthatOmust be an infinite set and therefore thatV (A)−BdryG(A) �= ∅. Moreover, sinceOis an�-class, no subgraph ofA, exceptA itself, can be a region ofG. Thus no automorphismof G can map a vertex of BdryG(A) to a vertex ofV (A)−BdryG(A), a contradiction. �

7.5. Peripheral regions containing exactly one odd-type class

The peripheral regions of a graph that contain exactly one odd-type class are of par-ticular interest because, as is shown by the following results, the circuit-decomposabilityof the whole graph is equivalent to the local circuit-decomposability of these regions, andbecause they have the surprising property that, regardless of whether they are locally circuit-decomposable or not, they are always locally cycle-decomposableto within a single edge.First note that by Proposition 7.13(ii) and the definition of an odd-type class (Definition

7.6), we have the following remark:

Remark 7.17. Every odd-type class of a graph is contained in some peripheral region thatcontains no other�-class of odd type.

This implies the following result.

Proposition 7.18.A graph G is circuit-decomposable if and only if every obstructive pe-ripheral regionofG that containsexactly oneodd-typeclass is locally circuit-decomposable.

Proof. Necessity follows from Theorem7.5. For the sufficiency it follows from Propo-sition 7.13(ii) that any maximal family of vertex-disjoint obstructive peripheral regionscontaining each at most one odd-type class, is also maximal when considered simply asa family of vertex-disjoint peripheral regions having only dominated rays. Thus the resultfollows by the implication(iv)⇒ (i) of Theorem 7.5. �

Proposition 7.19. Let A be a peripheral region of G containing exactly one odd-type classO. Then(G\e)/A ( = (G/A)\e) is cycle-decomposable for anye ∈ [O,A]G.

Proof. By Proposition7.13(i), there exists an edgee ∈ [O,A]G. Let us show thatG1 :=(G\e)/A is cycle-decomposable. Suppose the contrary. Then by Nash-Williams’s Theoremthere exists an odd cut[C,C]G1 such thatO ⊆ V (C). SinceA is of even degree inG1, we


may suppose without loss of generality thatA ∈ V (C), because otherwise, consider the cutinduced byC ∪ {A} instead ofC (by Lemma2.2 both are odd cuts). Hence,V (A) ∪O ⊆V (C), which implies that[C,C]G1 is also an odd cut ofG and thatC is contained in theperipheral regionA. This implies that one of the components ofC is also a peripheral regionofG. Thus, by Proposition 7.13(i),C contains an odd type class ofG, a contradiction to thefact thatO is the only odd type class ofG contained inA. �

The next result shows that peripheral regions containing exactly one odd-type class havea kind of extremal property with respect to all the regions of the graph.

Theorem 7.20.Let A be a peripheral region of G that contains a unique odd-type class O.Then any region B contained in A is either a locally cycle-decomposable(and hence even)region of G that is disjoint from O or a peripheral(and hence odd) region that contains O.

Proof. LetB ⊆ A be a region ofGand note thatO is either disjoint fromV (B) or containedin it. In the first case, by Proposition7.12,Bmust be a locally cycle-decomposable evenregion. In the second case, suppose by way of contradiction thatB is not a peripheralregion. SinceB ⊆ A, andA is peripheral,B does not contain odd cuts, and hence mustbe an even region because otherwise it would be peripheral. By Lemma 2.2, this impliesthat [A − B,A− B]G is an odd cut ofG, and therefore that at least one componentC ofA−B is an odd region that contains no odd-type class ofG. Since any odd region containedin a peripheral one is also peripheral, it follows thatC is peripheral, and this contradictsProposition 7.13(i). �

Corollary 7.21. Let A be a peripheral region having exactly one odd-type class. If A islocally circuit-decomposable, then one of the following holds:

(i) G/A has a decomposition into cycles and exactly oneqA-ray.

(ii) A contains an even region B of G such thatG/B has a decomposition into cycles andexactly twoq

B-rays.

Proof. LetObe the unique odd-type class ofG contained inAand� any decomposition ofG/A into circuits andq

A-rays. Sinceq

Ais a vertex of odd degree inG/A, we may suppose

without loss of generality that� contains exactly oneqA-ray. Now we have two cases to

consider.Case1: Some infinite fragment of� has a tail that is not�-dominated inG by any vertex

of O. Then there exists a regionB ⊆ A of G which is disjoint fromO and contains a tailof some fragment of�. Then by Proposition7.12,Bmust be a locally cycle-decomposableeven region, and it is easy to see that in addition,G/B satisfies the conditions of Lemma3.8. ThusG/B has a decomposition into cycles and exactly twoq

B-rays.

Case2: Each tail of each infinite fragment of� is �-dominated inG by some (and henceevery) vertex ofO. By Proposition 7.13(i), there exists an edgee ∈ [O,A]G. SinceA isconnected, there is aq

A-ray R in G/A such thate is the first edge ofR andR shares a

tail with the only ray of�. If we removeR from its�-shadow, the resulting subgraph iseulerian and locally finite and so is decomposable into circuits. Therefore(G/A)\R has a


circuit decomposition, and soG/A has a decomposition into circuits and exactly oneqA-ray,

namelyR. Denote byu the second vertex ofR. Since(G\e)/A admits a decomposition intocircuits and exactly oneu-ray, by Proposition7.19 it also admits a cycle decompositionand thus, by Proposition 3.7 forn = 1, admits a decomposition into cycles and exactly oneu-rayR′. Since{e} ∪E(R′) is the set of edges of either aq

A-ray or the edge-disjoint union

of aqA-ray and a cycle,G/A therefore admits a decomposition into cycles and exactly one

qA-ray as claimed. �

8. Removable subgraphs

In this section, we look for types of subgraphs which can be removed from the originalgraphwithout changing its circuit-decomposability or circuit-indecomposability. Hence theresults of this section are generalizations to the situation of circuit-decomposability of thefact that removing any finite eulerian subgraph from a (finite or infinite) graphG does notchange thecycle-decomposability ofG.

Proposition 8.1. Let H be a non-dominated eulerian subgraph ofG. If G is circuit-decomp-osable, then so isG\H , and conversely.

Proof. SinceH is locally finite by Corollary5.5 and eulerian, it is circuit-decomposableby Theorem 2.24. ThereforeG is circuit-decomposable ifG\H is circuit-decomposable.To prove the converse, assume thatG has a circuit decomposition�. Let K be the�-

shadow ofH. SinceG\K andH are circuit-decomposable, andK ⊇ H , it is sufficient toshow thatK\H also has such a decomposition. Being circuit-decomposable,K is eulerian,hence so isK\H becauseH is locally finite and eulerian. Let� = (�x)x∈V (G) be thetransition system ofG induced by�. For x ∈ V (K\H), let �′

x be the set of all pairs{e, e′} ∈ �x such thate, e′ ∈ E(K\H). SinceH is locally finite and eulerian, the numberof edges ofK\H incident withx but not belonging to any pair of�′

x is even. Therefore�′x

can be extended to a (full) transition system�′′x atx. Let �

′′ = (�′′x)x∈V (K\H). By Theorem

2.26,�′′ defines a decomposition�′′ of K\H into edge-traceable fragments.It will suffice to prove that each fragment of�′′ is circuit-decomposable. Therefore

suppose, by way of contradiction, that a fragmentM ∈ �′′ is not circuit-decomposable.Then, by Theorem 6.9, there is an odd regionA ofM such that every ray ofA is dominatedin M. SinceA is a subgraph of a edge-traceable graph, it is countable or finite, and henceby Theorem 2.10 has a rayless spanning treeT. Hence

V (A) ∩ V (H) is finite, (2)

because, by Lemma5.3, so isV (T )∩V (H). Note, however, thatAmust be infinite becauseotherwise the quotient graphM/(M − A) is a finite graph with exactly one vertex of odddegree, viz.q

M−A . HenceAmust contain a vertexu of infinite degree, because otherwiseA is infinite and locally finite and therefore contains a ray, and clearly no such ray can bedominated inA.Now let〈. . . , x−2, x−1, x0, x1, x2, . . .〉 be the 2-way infinite trail ofM induced by�′′. Let

U := {i; xi = u}. Note thatU is infinite and since[A,M−A]M is odd, itmust either contain


a least or a greatest element; without loss of generality suppose it has a smallest one, and letU = {i0, i1, . . .} with in < in+1. Let nA be a non-negative integer such thatxi ∈ V (A) foranyi� inA . Observe that for eachn�0, the sub-trailSn := 〈xin, xin+1, . . . , xin+1, xin+1+1〉contains three different edges that are incident withu. Since the transitions in� arise fromcircuits (i.e. from2-regular graphs), at least one of the�′′-transitions inSn does not belong to�. This means that there exists an indexjn with in < jn� in+1 such thaten := [xjn−1, xjn ]ande′n = [xjn, xjn+1] do not form a pair in�xjn . Thus, for eachn�nA, xjn belongs toV (A) ∩ V (H), and bothen ande′n are paired in�xjn with edges ofH. SinceH is locallyfinite, it follows thatV (H) ∩ V (A) is infinite, a contradiction to (2). �

Corollary 8.2. Let H be an eulerian subgraph of a circuit-decomposable graph G, and Dthe set of all vertices of G that dominate H. ThenG\H has a decomposition into circuits,D-rays and D-paths.

Proof. Without loss of generality suppose that each vertex adjacent to a vertex ofD is ofdegree 2 and the distance inG between any two vertices ofD is at least 3. If this is not thecase, subdivide each edge ofG (and consequently ofH) by two vertices. For eache ∈ E(G)incident to a vertex ofD, introduce a new rayRe at the vertex ofe that is not inD, such thatRe meetsG in that vertex only and such that theRe’s are pairwise disjoint. Now let

G := (G−D) ∪⋃e∈E(G)V (e)∩D �=∅

Re

and

H := (H −D) ∪⋃e∈E(H)V (e)∩D �=∅

Re.

It is easy to see thatH is an eulerian non-dominated subgraph ofG and thatG is circuit-decomposable. Hence, by Proposition8.1,G\H has a decomposition into circuits and sinceany such decomposition canonically induces a decomposition ofG\H into circuits,D-raysandD-paths, we are done.�

Intuitively, the last result together with Corollary 7.15 says that if one wishes to removesome subgraphH of a circuit-decomposable graphG in such a way that the remaining graphis still circuit-decomposable, a possible class of candidates forH are those subgraphs whichare not dominated inG by any vertex that is of odd type inG\H . Pursuing this idea furtherwe will extend Proposition 8.1 to the following class of subgraphs (see Theorem 8.6).

Definition 8.3. AsubgraphHofagraphG is said toberemovableif novertex thatdominatesH in G is of odd type inG\H .

An important situation where this concept is used is the case whereG\H is cycle-decomposable. Since in this caseG\H has no vertices of odd type (see Corollary7.8),His a removable subgraph ofG.


As is the case for the parity-type of vertices (cf. Lemma7.9), removability is preservedby the passage from a graph to some of its quotients:

Lemma 8.4. Let H be a subgraph of G and A a region of G. Then

H is removable inG �⇒ H/A is removable inG/A.

Moreover, if A contains all the vertices of G that dominate H, the converse is also true.

Proof. Straightforward from Lemma7.9, the definition of removable graphs and the factthat since[A,A]G is finite, the vertices that dominateH/A inG/A are exactly the verticesof A that dominateH in G. �

Definition 8.5. An eulerian-typegraphs is a graph that contains no odd-type vertex.

The following result which is a generalization of Theorem8.1, show that removablesubgraphs are “generalized” non-dominated subgraphs.

Theorem 8.6. Let H be a removable eulerian-type subgraph ofG. If G is circuit-decompos-able, then so isG\H , and conversely.

Proof. Sufficiency. Since all vertices ofH are of even type inH, by Corollary7.15,H iscircuit-decomposable and hence so isG.Necessity. LetK := G\H . We shall show that any obstructive peripheral region ofK

is locally circuit-decomposable. The circuit decomposability ofK then follows from theimplication(iii )⇒ (i) of Theorem 7.5.Let A be an obstructive peripheral region ofK. Denote byD the set of all vertices that

dominateH in G, and putDA := D ∩ V (A). LetO be the set of all vertices that are of oddtype inK, and letOA := O ∩ V (A).

Note that becauseH is removable the two setsD andO are disjoint and hence so areDA andOA. We now claim thatDA andOA can be separated by a finite cut, or, to be morespecific:

Claim. There exists an induced subgraphBofA such that[B,A−B]A is finite,DA ⊆ V (B)andOA ⊆ V (A− B).

Assuming for the moment that this claim is true, let us show that it implies the result.First note that since both[A,A]K and[B,A − B]A are finite, so are[B,K − B]K and

[A−B,K−(A−B)]K . By Corollary 8.2,K admits a decomposition� into circuits,D-raysandD-paths. Because of the finiteness of[A−B,K − (A−B)]K and the fact thatA−Bis an induced subgraph ofK,� may contain at most a finite number of fragments that meetA−B without being included in it. SinceD∩V (A−B) = ∅, the fragments of� containedinA−B are circuits, and hence their union is a circuit-decomposable subgraphL ofA−Bsuch that everyx ∈ V (A− B) is of even degree inK\L.


Now, putB := K−B. SinceV (B)∩O = ∅, Lemma7.9 implies that there is at most oneodd-type vertex inK/B, viz. q

B. Hence, if the vertexq

Bis not of odd type inK/B, then by

Corollary 7.15,K/B is circuit-decomposable. On the other hand, ifqBis of odd type, we

attach aqB-ray toK/B, obtaining a new graph with even type vertices only. By Corollary

7.15 this new graph is circuit-decomposable, and henceK/B itself has a decompositioninto circuits and exactly oneq

B-ray.

Hence in both cases we obtain thatB itself has a local circuit decomposition�1. LetM betheunionof the circuits of�1 that are contained inB. Since[B,B]K is finite, the vertices ofBmust all be of even degree inK\M, and hence inK\(M∪L) becauseL ⊆ A−B. Moreover,the vertices ofA−B are also of even degree inK\(M ∪L) because so are they inK\L andM ⊆ B. Thus the vertices ofA (= (A−B)∪B) are all of even degree inK\(M ∪L), andsinceM ∪ L is circuit-decomposable,A is therefore a locally circuit-decomposable regionof K, and we are done.

Proof of the Claim. By way of contradiction, assume that there is no finite cut ofA sep-aratingOA andDA. By Menger’s Theorem this implies thatA contains an infinite set ofedge-disjointOADA-paths. By Proposition 7.13(i),OA is composed of an odd, and hencefinite, number of odd-type classes ofK. Therefore one of these classes, sayU, is infinitelyedge-connected (inA) toDA. Since every odd-type class is also an�-class it follows that foreach vertexu ∈ U there is an infinite familyPu of edge-disjointuDA-paths inK. [A,A]Kbeing finite, we may suppose without loss of generality that the paths inPu belong toA.Denote byXu the vertices inDA that are endpoints of some path inPu.SinceDA ∩OA = ∅, and since any vertex which is infinitely edge-connected to an odd-

type vertex is also of odd type, no vertex ofDA can belong to infinitely many different pathsin Pu, implying that the setXu is infinite.We now apply Proposition 2.11 toA. Since every ray ofA is dominated inA it follows

that in both situations covered by Proposition 2.11 we have a vertexv ∈ V (A) and a familyLu of vXu-paths ofA which are vertex-disjoint except for their common endpointv. It iseasy to see that since each vertex inXu dominatesH in G, and sinceXu is infinite,v alsodominatesH in G; thusv ∈ DA. On the other hand, since clearlyv andu are infinitelyedge-connected inA, andu ∈ OA, v also belongs toOA, a contradiction. �

9. Having enough eligible rays, a necessary and sufficient condition

Recall that having enough rays and having enough non-dominated rays are respectively anecessary and a sufficient condition for a graph to be circuit-decomposable. It turns out thatin order to obtain a necessary and sufficient condition we have to define a new property ofthe rays of a graph which is a weakening of being non-dominated. It is tempting to considerremovable rays, but unfortunately there exist graphs that have enough removable rays butare not circuit-decomposable, see the two examples on the second line of Fig. 9. Since raysare never eulerian-type graphs, Theorem 8.6 does not fully apply here. Note however thatin those two examples of Fig. 9, each removable ray is dominated by its own origin. Hence,from the circuit decomposition point of view, odd-type vertices of removable subgraphs arebetter to be as far from dominating vertices as possible.


Fig. 8.

Definition 9.1. A subgraphH of a graphG is almost eulerianif the setOH of all the odd-type vertices ofH and the setDH of all the vertices that dominateH in G are separable bya finite cut ofG. In other word, if there exists a finite cut[A,A]G such thatDH ∈ V (A)andOH ∈ V (A).

It is a consequence of Theorem9.6 (see below) that having enough removable almosteulerian rays is a sufficient condition for circuit-decomposability. However this is not theproperty we are looking for, because there exist graphs that are circuit-decomposable butdo not have enough removable almost eulerian rays (an example is shown in Fig. 8 wherethe only removable ray is the one going to the left). Hence we have to consider a larger classof rays.

Definition 9.2. A rayRofG is said to beeligible if it is contained in a locally finite almosteulerian removable subgraph.

Observe that removable almost eulerian rays (and hence non-dominated rays) are eligiblebut the converse is not true in general. However, as is shown by the next result, there isonly one other case to consider: the “removable 2-rays” (a2-ray being a graph that is theunion of two edge-disjoint rays having the same origin). Note also that in each exampleof the second line of Fig.9, there are enough rays that are contained in an almost eulerianremovable subgraph. Hence the local finiteness condition in the definition of eligibilitycannot be omitted if one want to characterize the circuit-decomposability.Given any locally finite circuit-decomposable graphG and anyx-rayR of G, the graph

G\R is still locally finite and has exactly one vertex of odd degree (viz.x). This impliesthatG\R contains anx-rayR′. Since the subgraphS := R ∪R′ of G is a 2-ray, it followsfrom Theorem 2.24 thatG\S is circuit-decomposable. Hence, in a locally finite circuit-


decomposable graph any rayR is contained in a 2-ray that is a member of a decompositionof G into circuits and 2-rays. Since any such decomposition trivially induces a circuit de-composition, one might say that to construct a circuit decomposition of a locally finitecircuit-decomposable graph, each ray is “usable”. The next Proposition shows that in arbi-trary circuit-decomposable graphs, each eligible ray is “usable”.

Proposition 9.3. Let R be an eligible u-ray of a circuit-decomposable graph G. Then thereexists a u-rayR′ contained inG\R such thatG\(R ∪R′) is still circuit-decomposable.

Proof. Let H be a locally finite almost eulerian removable subgraph ofG that containsR. LetOH be the set of all the odd-type vertices ofH, DH the set of all the vertices thatdominateH in G, and[A,A]G a finite cut ofG such thatDH ∈ V (A) andOH ∈ v(A).For every vertexx ∈ OH , define two new raysR1

x andR2x such thatR

1x ,R

2x andGpairwise

intersect inx only and such thatRix andRj

x′ are vertex-disjoint ifx �= x′. Let

G := G ∪⋃x∈OH

(R1x ∪ R2

x) and H := H ∪⋃x∈OH

R1x.

SinceH is locally finite, each vertex ofOH is of odd degree inH. This implies thatH isa locally finite eulerian graph, and hence an eulerian-type graph. Let us show thatH is alsoa removable subgraph ofG. SinceV (Rix) ∩ V (A) = ∅ for anyx ∈ OH andi ∈ {1,2}, thefinite cut [A,A]G is also a cut ofG. Hence(G\H )/(G − A) = (G\H)/(G − A), and ittherefore follows from Lemma7.9 that any vertex ofAhas same parity type inG\H and inG\H . By construction, the set of vertices that dominateH in G is exactlyDH . This impliesthat each vertex ofDH is of even-type inG\H becauseDH ⊆ V (A) and becauseH is aremovable subgraph ofG. ThusH is removable inG.SinceG is clearly circuit-decomposable, by Theorem 8.6, there exists a circuit decom-

position�G\H of G\H . Let K be�

G\H -shadow of⋃x∈OH R

2x , andK := K ∩ G. It is

easy to see thatK ∪ H is a locally finite eulerian subgraph ofG. Hence(K ∪ H)\R islocally finite andu is its only vertex of odd degree. This implies that there exists anu-rayR′ contained in(K ∪H)\R. Since(K ∪H)\(R ∪R′) is still a locally finite eulerian graph,by Theorem 2.24, it has a circuit decomposition�(K∪H)\(R ∪R′) which, together with allthe fragments of�

G\H that are not contained inK, form a circuit decomposition ofG\(R ∪R′). �

Proposition 9.4. LetA be an odd region ofG.Then the following statements are equivalent.

(i) A contains an eligible ray;(ii) A∪ [A,A]G contains some almost eulerian removable ray or A contains some remov-

able2-ray;(iii) A contains a non-dominated ray or a region B such thatG/B has a decomposition into

cycles and exactly one or exactly twoqB-rays.

Proof. (ii ) ⇒ (i) is evident because an almost eulerian ray and a 2-ray are always locallyfinite and almost eulerian subgraphs.


(iii )⇒ (ii ). SupposeAcontains no non-dominated rays, and letBbea region that satisfiescondition (iii). By the remark that follows thedefinitionof a removablegraph (Definition8.3)and by the converse part of Lemma 8.4, we are done ifG/B has a decomposition into cyclesand exactly oneq

B-ray. Hence suppose thatG/B has a decomposition� into cycles and

exactly twoqB-rays.B being a connected subgraph ofG/B, q

Bis not a cut-vertex ofG/B.

Hence, inB (= G/B − qB), there exists a vertexv and two edge-disjointv-raysR1 and

R2 that have the tails of the twoqB-rays of�. By Lemma 2.30, the set{R1, R2} can be

extended to a decomposition ofG/B in which all the other fragments are cycles. ThusG/B − (R1 ∪ R2) is cycle-decomposable. SinceqB �∈ V (R1 ∪ R2), again by the remarkfollowing Definition 8.3 and Lemma 8.4, we have thatR1∪R2 is connected and removablein bothG/B andG. ThusR1 andR2 are eligible inG.(i) ⇒ (iii ). Suppose thatA contains dominated rays only. By Proposition 7.13(i),A

contains an odd number of odd-type class, sayO1,O2, . . . , O2n+1, (n ∈ �). LetRbe anyeligible ray ofG contained inA, anduany of its dominating vertex. Since[A,A]G is finite,u ∈ V (A). Now letH ⊇ R be any removable subgraph ofGas given by Definition 9.2, and[C,C]G be a finite cut ofG such thatV (C) containsDH (the set of vertices that dominateH in G) andV (A) containsOH (the set of vertices that are of odd degree inH). Observethat, by Lemma 2.2,[A ∩ C,A ∩ C]G is finite. We now have to consider two cases.Case1:u �∈ ⋃2n+1

i=1 Oi . Then, by Lemma2.3, there exists a regionBofG that is containedinA∩C, that containsu, and that is disjoint from⋃2n+1

i=1 Oi . ByTheorem7.20,B is a locallycycle-decomposable even region. Hence,G/B is cycle-decomposable. Moreover, sinceudominatesR, B contains a tail ofR. LetQ be any such ray, andx be its origin.Since no vertex ofH/B except possiblyqB is of odd degree inH/B, the latter is therefore

locally finite and almost eulerian. Moreover,H/B is, by Lemma 8.4, removable inG/B.SinceQ is contained inH/B, it is therefore an eligible ray ofG/B. Thus, it follows fromProposition 9.3 that there exists anx-rayQ′ such thatG/B has a decomposition�G/B intocircuits and exactly twox-rays, the twox-rays beingQ andQ′. Applying Theorem 2.24,let�Q∪Q′ be a circuit decomposition ofQ ∪Q′. Thus� := �G/B\{Q,Q′} ∪ �Q∪Q′ is a

circuit decomposition ofG/B.If � contains at least a double-ray, then, by Lemma 3.8,G/B has a decomposition into

circuits and exactly twoqB -rays, as desired.If � is a cycle decomposition, thenQandQ′ are the only infinite fragments of�G/B , and

the existence of a decomposition ofG/B into circuits and exactly twoqB -rays is this time aconsequence of Lemma 2.31 (replacingG, v, and� byG/B, qB and�G/B , respectively).Case2: u ∈ Oj , for somej. Then, by Lemma 2.3, there exists a regionDOj of G that

is contained inA ∩ C, that containsu (and henceOj ), and that is disjoint from⋃i �=j Oi .By Theorem 7.20,DOj is a peripheral region having exactly one odd-type class, vizOj .SinceDOj ⊆ A, without loss of generality we may suppose thatDOj = A, and thereforethatOH ⊆ V (A). By Corollary 7.21 and Lemma 8.4, to finish the proof, we only have toshow thatA is locally circuit-decomposable.By Proposition 7.13(i), there existsv ∈ Oj ∩BdryG(A) and hence somevA-edgee. Let

M be a connected graph (disjoint fromG/A) that has a cycle decomposition as well as adecomposition into cycles and exactly one rayS. LetQ be a ray disjoint fromG/A andM,


and form a new graphK from the union ofG/A,M andQ by identifying the verticesv andqAof G/A with the origins ofSandQ, respectively.K is circuit-decomposable because

M ∪ e ∪ Q obviously is circuit-decomposable, and(G/A)\e is cycle-decomposable byProposition7.19. We leave it to the reader to show thatOj is still the only odd-type classof K, and that(H/A) ∪M and(H/A) ∪M ∪Q are both removable inK (here one usesthatv already dominatesH/A in G/A). SinceOH ⊆ A, one ofH/A and(H/A) ∪Q iseulerian, we therefore have by Theorem 8.6 that one of

K\((H/A) ∪M) andK\((H/A) ∪M ∪Q)is still circuit-decomposable. In either case this implies that(G/A)\(H/A) has a decompo-sition into circuits andq

A-rays and therefore, sinceH/A is locally finite and has no vertex

of odd degree except possiblyqA, thatG/A itself admits a decomposition into circuits and

qA-rays. ThusA is locally circuit-decomposable, and we are done.�

The following result is our main theorem.

Theorem 9.5. A graph is circuit-decomposable if and only if it has enough eligible rays.

Proof. Thenecessity is adirect consequenceofProposition7.18,Corollary7.21, andPropo-sition 9.4. For the sufficiency, by Proposition 7.18, we have to show that each obstructiveperipheral region that contains exactly one odd-type class is locally circuit-decomposable.LetG be a graph andA any such region,O the only odd-type class inA, andK the graphobtained fromG/A by attaching a new rayR0 to the vertexqA . To finish the proof, let usshow thatK is circuit-decomposable. Lemma 7.9, applied toA as induced subgraph ofG,and again toAas induced subgraph ofK shows that the type of the vertices ofA is the samein G, G/A, andK. Thus,O is also the unique odd-type class ofK. Moreover, sinceG hasenough eligible rays and sinceR0 is non-dominated (and hence eligible) inK, Lemma 8.4implies thatK has enough eligible rays.Case1: There exists a regionB of G contained inA such thatG/B has a decomposition

into cycles and exactly oneqB-ray (in particular,B is locally circuit-decomposable). ThenB

must be an odd region and therefore, by Theorem 7.20, a peripheral region ofG (and hencealso ofK). Moreover, sinceK has exactly one odd-type class, it follows from Proposition7.13(i) thatK contains no peripheral region that is disjoint fromB. Hence{B} is a maximalfamily of vertex-disjoint obstructive peripheral region, implying, by Theorem 7.5((iv) ⇒(i)), K is circuit-decomposable.Case2: There is no region having the properties of Case 1. Then byProposition 9.4((i)⇒

(iii )) there is a non-empty family(Bi)i∈I of vertex-disjoint regions ofG that are containedin A such that eachG/Bi has a decomposition�i into cycles and exactly twoq

Bi-rays.

Suppose that(Bi)i∈I is maximal with respect to these properties, and note that eachBimust be an even region and hence, by Theorem 7.20, is disjoint fromO. We claim that inG(and hence also inK),O cannot be separated from

⋃i∈I Bi by a finite cut. Indeed, if there

exists a regionC of G that containsO and is disjoint from⋃i∈I Bi , then the component of

C − A that containsO is, by Theorem 7.20, a peripheral region ofG. SinceG has enougheligible rays, Proposition 9.4((i) ⇒ (iii )) implies that either there exists a region having


the properties of Case 1, or the family(Bi)i∈I is not maximal; in either case we reach acontradiction.For eachi ∈ I letLi be the�i-shadow of some finite connected subgraph ofG/Bi that

contains all the edges incident to the vertexqBiand in whichq

Biis not a cut-vertex. Such

anLi exists because,Bi being connected,qBiis not a cut-vertex ofG/Bi in the first place.

Then letKi be the graph obtained from(G/Bi)\Li by deleting isolated vertices. Observethat these graphs have the following properties:

• qBidoes not belong toKi (or equivalentlyKi ⊆ Bi);

• Li contains at least two edge-disjoint rays, namely the twoqBi-rays of�i ;

• Ki is cycle-decomposable;• Li − qBi is connected and locally finite.Now considerL := K\(⋃i∈I Ki). Note that

E(L ∩ Bi) ∪ [qBi, Bi]G/Bi = E(Li). (3)

To finish the proof, it suffices to show thatL has a circuit decomposition. Suppose thecontrary. By Proposition7.18, there is an obstructive peripheral regionD of L that con-tains exactly one odd-type class of the graphL, sayOD, and which is not locally circuit-decomposable. By (3), the vertices of eachBi have the same degree inL as inLi , and sinceLi is the shadow of a finite subgraph ofG/Bi , eachx ∈ V (Bi) has even degree inLi , andhence inL. Therefore none of the vertices ofBi , i ∈ I , belongs toOD.Note that[Li−qBi , Li − qBi ]L is finite because it is equal to[Bi, Bi]K . Sincemoreover,

Li − qBi is locally finite, each ray contained inLi is non-dominated inL. Hence each rayof each�i (i ∈ I ) induces a non-dominated ray inL, and none of these has a tail inD.Therefore by (3) and the connectedness ofLi − qBi , it follows that if a vertex ofBi belongstoD, then at least one edge ofBi belongs to[D,D]L, for anyi ∈ I . Since[D,D]L is finite,so isV (D) ∩ (⋃i∈I V (Bi)). Moreover,D being an obstructive peripheral region ofL, atmost a finite initial segment of the rayR+

0 := [qA,G/A− qA]K ∪ R0 belongs toD. Thus,

V (D) ∩(V (R+

0 ) ∪⋃i∈IV (Bi)

)is finite.

By Lemma2.3,V (R+0 )∪

⋃i∈I V (Bi) is therefore separable fromOD by a finite cut inL. Let

N ⊆ D beany regionofL that contains the verticesofOD but novertexofV (R+0 )∪

⋃i∈I Bi ,

and observe thatN is also contained inA (= K − V (R+0 )) and that

[N,K −N ]K = [N,L−N ]L. (4)

Moreover,N cannot containO because, as proved earlier,O is not separable inK from⋃i∈I V (Bi) by a finite cut. SinceA is peripheral inK and does not containO, by Theorem

7.20,N is therefore an even region ofK. On the other hand, again by Theorem 7.20,N is anodd region ofL because it is contained inD (which is peripheral inL) and containsOD, acontradiction to (4). �


Theorem9.5 can be stated in this apparently stronger form.

Theorem 9.6. A graph G is circuit-decomposable if and only if every peripheral regioncontains an eligible ray.

Proof. Theorem9.5, Proposition 7.4 and the fact that a non-dominated ray is alwayseligible. �

In the remainder of this section we will show that “most” of the ends of a graph containeligible rays. This will imply that eulerian graphs that are not circuit-decomposable musthave a very particular structure. The next two propositions will make this statement moreexplicit.

Proposition 9.7. Every end of G that is of�-multiplicity> 2 contains some eligible ray.

Proposition 9.8. Every end of G that is of�-multiplicity > 1 and is�-dominated by anodd-type vertex of the graph contains some eligible ray.

We first prove Proposition9.8 as it will be used in the proof of Proposition 9.7.

Proof of Proposition 9.8. Let � be such an end,O the odd-type class whose members�-dominate�, andAbe a peripheral region that containsOand no other odd-type class ofG. ByRemark 7.17 such anA exists. Clearly, each ray of� has a tail inA. By Proposition 7.13(i),there existsu ∈ O ∩BdryG(A); let e ∈ [u,A]G. Note that by Proposition 7.19 there existsa cycle decomposition� of the graph(G\e)/A. Since the�-multiplicity of � is greater than1, then by Theorem 3.3,(G\e)/A has a decomposition�′ into cycles and exactly oneu-ray,sayR. Observe thatR ∪ e is either a ray or the edge-disjoint union of a tail ofRand a cycle.Without loss of generality we may suppose thatR ∪ e is a ray because in the other possiblecase, we can take that specific tail ofR instead ofRand then add the extra cycle so obtainedto�′.Now, since�′\{R} is a cycle decomposition of(G/A)\(R ∪ e), it follows from Corol-

lary 7.8 that(G/A)\(R ∪ e) has no vertex of odd-type. HenceR ∪ e is removable inG/Awhich, by Lemma 8.4, implies that the raySof G induced by the edges ofR ∪ e is remov-able inG. Since moreover the origin ofS is in A, S is therefore an eligible ray containedin �. �

Proof of Proposition 9.7.Without loss of generality assumeG to be connected. By wayof contradiction, suppose that there exists an end� of �-multiplicity > 2 that contains noeligible ray. Since� must be dominated, the setX that consists of all the vertices ofG that�-dominate� in G is non-empty and hence is an�-class. Moreover, by Proposition 9.8,Xis an even-type class. Now there are two cases to consider.Case1:X is contained in some peripheral regionA of G. Since by Proposition 7.13(i),A

contains only finitely many odd-type classes, it follows from Lemma 2.3 that there existsa regionB ⊆ A which containsX but no odd-type class ofG. Note that each ray of�has a tail inB and that by Proposition 7.12,B is locally cycle-decomposable. HenceG/B


is cycle-decomposable which implies by Theorem3.3 that given anyx ∈ X, G/B hasa decomposition into cycles and exactly twox-raysR1, R2 ∈ �. By Lemma 2.30 we maysuppose that neitherR1 norR2 contains the vertexqB . This implies thatR1∪R2 is removablein G/B and hence, by Lemma 8.4, also removable inG. Thus bothR1 andR2 are eligiblerays.Case2:X is not contained in anyperipheral region ofG. Letubeany vertex that dominates

� in G (henceu ∈ X), and considerH := R1 ∪ R2 ∪ R3 ∪ R12 ∪ R13 ∪⋃i∈� Pi , whereR1, R2, R3 are three edge-disjointu-rays of�, andR12 andR13 areu-rays of� such thatR1j meets bothR1 andRj infinitely often,j = 2,3, and(Pi)i∈� is an infinite family ofuR1-paths having onlyu in common. Let(Ai)i∈I be a maximal family of (vertex-)disjointodd regions ofG that are (vertex-)disjoint fromH, and putA := ⋃

i∈I Ai .Construct a new (multi-)graphG as follows: add a new vertexq toGand� newqy-edges

for each odd type vertexy of G that is not inA, and then identify all the vertices ofAwithq (deleting loops). The vertex ofG resulting from this identification will be denoted byq.Note that inG, the odd type vertices ofG have either been identified withq or are joinedto q by infinitely many edges.We claim that every odd region ofG containsq. By way of contradiction letB be an odd

region ofG such thatq �∈ V (B). Being contained inG−A (= G−q),B is also anodd regionof G. SinceB cannot contain any odd type vertex ofG, it follows from Proposition 7.13(i)that B contains no peripheral region ofG. By Proposition 7.2,B therefore contains an�-stratifying sequence(Bn)n∈� of odd regions ofG. Since each ray contained inH isdominated (and hence�-dominated) inG, by Corollary 2.18 there exists ann0 ∈ � suchthatV (H) ∩ V (Bn0) = ∅. This is a contradiction to the maximality of the family(Ai)i∈IbecauseBn0 ⊆ B, andB is disjoint fromA (= ⋃

i∈I Ai). Thus every odd region ofGcontainsq, as claimed.It follows from this claim thatG has no odd cut, and hence is cycle-decomposable by

Nash-Williams’s Theorem. SinceH ⊆ G, u �-dominates� in G, and by Theorem 3.3(applied toG and to the end� of G that contains the rayR1), G has a decomposition� intocycles and exactly twou-rays belonging to� (RandR′, say). Sinceu �-dominates� also inG− q, there exist two edge-disjointu-rays ofG that have tails in{R,R′} and do not containthe vertexq. Hence, by Lemma 2.30, wemay suppose that� has been so chosen that neitherRnorR′ containsq, or in other words thatRandR′ belong toG.Abeing disjoint fromH,RandR′, it is easy to see thatRandR′ are both end-equivalent toR1 inG. HenceR,R′ ∈ �.PutQ := R ∪R′. ClearlyQ is a locally finite eulerian graph ofG. To complete the proof,

let us show thatQ is removable inG. By way of contradiction, suppose that there is a vertexx that dominatesQ in G and is of odd type inG\Q. Note thatx belongs toX, and hence isof even type inG; also, being an�-dominating vertex of each ray inH, x does not belong toanyAi . LetOx be the�-class ofx inG\Q, and applying Remark 7.17, choose a peripheralregionC of G\Q such that the only odd type class ofG\Q contained inC isOx .Since every vertex that�-dominatesQ in G belongs toX and hence is of even type inG,

it follows from Proposition 7.10 that each vertex that is of odd type inG is also of odd typein G\Q. Sincex ∈ X (which is an even type class ofG), no vertex that is of odd type inGmay belongOx . ThusC contains no vertex that is of odd type inG.


TheAi ’s are odd regions ofG\Q because they are odd regions ofGandV (Ai)∩V (Q) =∅. Moreover, ifV (Ai) ∩ V (C) �= ∅ then,Ai being connected,

eitherAi ⊆ C or E(Ai) ∩ [C,C]G\Q �= ∅.The first possibility implies that[

Ai, (G\Q)− Ai]G\Q ∩ [C,C]G\Q �= ∅

becauseC is peripheral inG\Q. On the other hand, theAi ’s being disjoint, each edgee ∈ [C,C]G\Q can belong toE(Ai) for at most onei ∈ I , or to [Ai, (G\Q)−Ai]G\Q forat most twoi ∈ I . Thus the setJ of all the indicesi ∈ I such thatV (Ai) ∩ V (C) �= ∅ isfinite.If J = ∅, putC′ := C, otherwise letK := ⋃

j∈J Aj andC′ be the component ofC−Kthat containsx (recall thatx does not belong to anyAi). Since[K,K]G\Q is clearly finite,so is[C −K,C −K]G\Q by Lemma2.2. HenceC′ is a region ofG\Q. Moreover, sinceOx is the only odd type class ofG\Q in C, it follows from Theorem 7.20 (withG,O, B,Areplaced byG\Q,Ox , C′, C, respectively) thatC′ is an odd region ofG\Q.The regionC′ is disjoint fromA (= ⋃

i∈I Ai). Therefore in the passage fromG to G,no vertex ofC is identified withq, so thatC′ is contained inG− q. Moreover, every edgeof G incident with a vertex ofC′ is also an edge ofG (possibly with one of its endpointsidentified withq). On the other hand, being a subset ofC, C′ does not contain any vertexthat is of odd type inG. This means that in the passage fromG to G, no edges are added atany vertex ofC′. Hence the cuts induced byC′ in G\Q andG\Q coincide, which meansthatC′ is an odd region ofG\Q, a contradiction to Nash-Williams’s Theorem becauseQ = R ∪R′ and� − {R,R′} is a cycle decomposition ofG\(R ∪R′). �

From Proposition 9.7 and Remark 2.4, we obtain the following corollary.

Corollary 9.9. An end which admits infinitely many dominating vertices contains an eli-gible ray.

Moreover, it follows from Propositions9.7 and 9.8 that in a graph which is not circuit-decomposable there must exist a peripheral region in which ends are very “thin”. We gen-eralize this idea in Proposition 9.11, which is based on the following definition.

Definition 9.10. An end� of a graphG is said to bethreadlikeif it has�-multiplicity 1 andat least two dominating vertices.

By Remark2.4, a threadlike end has at most finitely many dominating vertices.The next proposition says that in a region that has no eligible ray, “almost” all ends are

threadlike and the exceptions are “almost threadlike” in the sense that they satisfy only oneof the two defining conditions of threadlike ends. We shall use the following notation: fora vertexx ∈ V (G) denote byDx the set of all ends ofG that are�-dominated byx in G.

Proposition 9.11. Let A be a peripheral region of G that contains no eligible ray. Then allrays in A are dominated, and given any vertexx ∈ A of infinite degree, either


(i) x is of odd type and all ends inDx are threadlikeor(ii) x is of even type and all but at most one end inDx are threadlike, and the exceptional

end� (if it exists) has one of the following properties:(1) � has�-multiplicity 1 and exactly one dominating vertex;(2) � has�-multiplicity 2 and at least two dominating vertices.

Proof. It is a direct consequence of the definition of eligibility that all rays inA are domi-nated.Case(i): x is of odd type. Let� ∈ Dx . Then by Proposition9.8,� is of �-multiplicity 1.

Moreover, by exactly the same argument as in the proof of Proposition 9.8 except thatTheorem 3.3 is replaced by Proposition 3.6, it follows easily that� cannot be dominated byexactly one vertex. Thus� is threadlike.Case(ii): x is of even type. Suppose there are two distinct ends�, �′ in Dx that are not

threadlike.As in the proof of Proposition 9.7 (case 1), choose a locally cycle-decomposable region

B ofG that is contained inA and that containsx but no odd-type class ofG. Thus every endinDx has rays inB. Letube any vertex that dominates� inG. Clearlyu is inB. Observe thatu is infinitely edge-connected tox and hence�-dominates�′ in G (and therefore inG/B),and that for any rayR ∈ �′, ustill dominates� inG\R and therefore also in(G/B)\(R/B).Now apply either Theorem 3.3 or Proposition 3.6 in order to construct a decomposition ofG/B into cycles and exactly oneu-rayR′ ∈ �′. Observe thatG := (G/B)\R′ is cycle-decomposable and that,� and�′ being different ends ofG, the set� of all rays of� that arecontained inG is an end inG. It is easy to see that� is not threadlike and still dominatedby u in G.Again, apply either Theorem 3.3 or Proposition 3.6 to construct a decomposition ofG

into cycles and exactly oneu-ray R ∈ �. ClearlyD := R′ ∪ R is a removable 2-ray inG/B becauseD is locally finite and eulerian and(G/B)\D is cycle-decomposable. ByLemma 8.4,D is also removable inG. R′ andR are therefore eligible rays ofG containedin A, a contradiction to the hypothesis of the proposition.To finish the proof consider a non-threadlike end� that has rays inA. By Proposition 9.7,

� has�-multiplicity �2. By an argument that follows verbatim the proof of Proposition 9.7(case 1) except that we use Proposition 3.6 instead of Theorem 3.3, one may once againeasily show that every end of�-multiplicity = 2 that has rays inAmust be dominated bymore than one vertex ofG. This implies that one of properties (1) and (2) holds for the end�. �

Proposition 9.12.Agraph having atmost one vertex of infinite degree is circuit-decompos-able if and only if it has enough rays.

Proof. The necessity has already been proved in Proposition4.2. For the sufficiency, sup-poseG is a counterexample. Having enough rays,G cannot contain vertices of odd degree,but by Corollary 7.15, it contains a vertex of odd-type. HenceGcontains exactly one vertexof infinite degree, and that vertex is of odd-type inG. Denote that vertex byx. Apply Theo-rem9.6 to choose a peripheral regionA that contains no eligible rays. ByProposition 7.13(i),


Fig. 9. Peripheral regions that are not locally circuit-decomposable.

x belongs toA. Sincex is the only vertex of infinite degree, no end ofA is threadlike, butxmust dominate (and therefore�-dominate) each ray ofA. Thus, by Proposition9.11, theodd regionAmust be rayless, contradicting the hypothesis.�

It follows fromTheorem 9.6, Proposition 7.13(i) and Proposition 9.11 that a graph whichis not circuit-decomposable, must contain an odd-type vertex that�-dominates threadlikeends only. Moreover, those results together with Proposition 7.13(ii) say that if a graph isnot circuit-decomposable, then it must contain a peripheral region which has exactly oneodd-type class, which is in some sense obtained by identifying vertices of some of the fourtypes of odd regions shown in Fig. 9. Each of these four regions is obviously peripheral,contains exactly one odd-type class and is not locally circuit-decomposable.

Acknowledgments

This paper is part of a Ph.D. thesis written under the supervision of Gert Sabidussi. Theauthor wishes to thank him for his support and comments throughout this work.


References

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127.[9] H.A. Jung, Wurzelbäume und unendliche Wege in Graphen, Math. Nachr. 41 (1969) 1–22.[11] F. Laviolette, Decompositions of infinite graphs: Part I—bond-faithful decompositions, Preprint, Université

de Montréal, 1996.[12] F. Laviolette, N. Polat, Spanning trees of countable graphs omitting sets of dominated ends, Discrete Math.

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Further reading

[10] F. Laviolette, Decomposition of infinite eulerian graphs with a small number of vertices of infinite degree,Discrete Math. 130 (1994) 83–87.

[17] C. Thomassen, Infinite graphs, in: Selected Topics in Graph Theory 2, Academic Press, London, 1983, pp.129–160.

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