Matroids, graphs in surfaces, and theTutte polynomial
2016 International Workshop on Structure in Graphs andMatroids
Iain Moffatt and Ben Smith
Royal Holloway, University of London
Eindhoven, 29th July 2016
13
Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
Overview
I Introduce matroidal analogues of various notions ofembedded graphs.
I Introduce by applications to the theory of the Tuttepolynomial:1. Extensions of the Tutte polynomial to graphs in
surfaces.2. Incomplete aspects of the theory.3. matroid model.4. Topological graphs ↔ matroid models
13
2 Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
A review of the Tutte polynomial
The Tutte polynomial, T(G;x,y)
Polynomial valued graph invariant, T : Graphs→ Z[x,y].
I Importance due to applications / combinatorial info.(colourings, flows, orientations, codes, Sandpile model,Potts & Ising models (statistical physics), QFT, Jones &homflypt polynomials (knot theory), ...)
Definition (deletion-contraction)
T(G;x,y) =
1 if G edgelessxT(G/e) if e a bridgeyT(G\e) if e a loopT(G\e) + T(G/e) otherwise
13
2 Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
A review of the Tutte polynomial
State sum formulation (T(G) is well-defined)
T(G) =∑A⊆E
(x− 1)r(G)−r(A)(y − 1)|A|−r(A)
where r(A) = #verts.−#cpts. of (V,A) = rank of A .
I T is defined for matroids (e.g., r= rank function).I T(C(G)) = T(G), where C(G) is cycle matroidI Matroids often ‘complete’ graph results (e.g.duality)
13
Tutte polynomial
3 Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
Graphs in surfaces
I Plane graph - drawn on a sphere, edges don’tmeet, faces are disks.
I Embedded graph = graph in surface - drawn onsurface, edges don’t meet.
I Cellularly embedded graph - drawn on surface,faces are disks.
13
Tutte polynomial
4 Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
A Topological Tutte polynomial
The Bollobás-Riordan-Krushkal polynomialK(G;x,y,a,b) :=
∑A⊆E(G)
xr(G)−r(A)y|A|−r(A)aγ(A)bγ∗(Ac)
γ(A) := Euler genus of nbhd. of subgraph of G on Aγ∗(Ac) := Euler genus of nbhd. of subgraph of G∗ on Ac
I T(G;x,y) = K(G;x− 1,y − 1,1,1)
I G plane graph =⇒ T(G;x,y) = K(G;x−1,y−1,a,b).
13
Tutte polynomial
4 Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
A Topological Tutte polynomial
I Deletion-contraction definition of the topologicalTutte polynomial:
13
Tutte polynomial
4 Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
A Topological Tutte polynomial
I Deletion-contraction definition of the topologicalTutte polynomial:
I No (full) recursive definition.
13
Tutte polynomial
4 Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
A Topological Tutte polynomial
I Deletion-contraction definition of the topologicalTutte polynomial:
I No (full) recursive definition.I =⇒ cell. embedded graphs are not the correctframework for the topological Tutte polynomial!
I What is the correct framework?
13
Tutte polynomial
Topologicalextensions
5 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
Look to matroids
I Why does deletion-contraction fail?
wants ribbon graph contraction
wants graph contraction
wants deletion as contraction in dual
¿ contract ?
I Exponents demand incompatible notions ofdeletion and contraction....
13
Tutte polynomial
Topologicalextensions
5 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
Look to matroids
I Why does deletion-contraction fail?
wants ribbon graph contraction
wants graph contraction
wants deletion as contraction in dual
¿ contract ?
Cycle matroid, C(G)
Bond matroid, B(G*)
Delta-matroid, D(G)
I Exponents demand incompatible notions ofdeletion and contraction....
I ...but these are provided by various matroids.
13
Tutte polynomial
Topologicalextensions
6 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
Delta-matroids
Symmetric Exchange Axiom (SEA): ∀X,Y ∈ F , if ∃u ∈ X4Y,then ∃v ∈ X4Y such that X4{u,v} ∈ F .
matroids (via bases)M = (E,B)
I B 6= ∅, subsets of EI B satisfies SEAI X,Y ∈ B =⇒ |X| = |Y|
Cycle matroid (trees)
M(G) = (E, {{2}, {3}})
delta-matroidsM = (E,F)
I F 6= ∅, subsets of EI F satisfies SEAI X,Y ∈ F =⇒ |X| = |Y|
∆-matroid (quasi-trees)
D(G) = (E, {{1,2,3}, {2}, {3}})
I Dmin = (E, {smallest sets}) a matroidI Dmax = (E, {biggest sets}) a matroidI D(G)min = C(G)I D(G)max = B(G∗) = (C(G∗))∗
13
Tutte polynomial
Topologicalextensions
7 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
(matroid, delta-matroid, matroid)
I Associate triple to embedded graph:
I Generally, consider triples
(M,D,N) of (matroid, delta-matroid, matroid)
I Deletion & contraction:
(M,D,N)\e := (M\e,D\e,N\e), (M,D,N)/e := (M/e,D/e,N/e)
I Important observation: different actions of deletioncontraction,
(Dmin)/e 6= (D/e)min, (D\e)max 6= (Dmax\e).
(So we have more than the delta-matroid.)
13
Tutte polynomial
Topologicalextensions
8 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
Strong maps and matroid perspectives
I There is structure we are not seeing.I Not all (graphic) triples can arise as minors of
(B(G∗),D(G),C(G)),I e.g., 12 triples (M,D,N) on 1 element, only 5 arise.I =⇒ missing conditions.
Matroid perspectivesA matroid perspective, is a pair of matroids (M,N) overE such that1. ⇐⇒ every circuit of M is union of circuits of N2. ⇐⇒ every flat of N is a flat of M,3. ⇐⇒ rM(B)− rM(A) ≥ rN(B)− rN(A) when A ⊆ B ⊆ E4. ⇐⇒ M = H\A and N = H/A, for some H on E t A.
I Examples of matroid perspectivesI (B(G∗),C(G))I (C(G),C(H)) where H from G by identifying verticesI (Dmax,Dmin) where D a delta-matroid
13
Tutte polynomial
Topologicalextensions
9 A matroidal setting
Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
∆-perspectives
∆-perspectivesAn ∆-perspective is a triple (M,D,N) such that
1. M and N are matroids, and D is adelta-matroid over the same set,
2. (M,Dmax) is a matroid perspective3. (Dmin,N) is a matroid perspective
I Example: (B(G∗),D(G),C(G)) is a ∆-perspective.
TheoremIf (M,D,N) is an ∆-perspective, then so are (M,D,N)\eand (M,D,N)/e.
(M,D,N) from cell. embed. graph ; its minors are.
13
Tutte polynomial
Topologicalextensions
A matroidal setting
10 Matroidpolynomials
Graphicalanalogues
Unifying TopologicalTutte polynomials
‘Tutte polynomial’ of perspectives
I There is a canonical way to construct ‘Tuttepolynomials’ of objects (via Hopf algebras).
Definition: Tutte polynomial of (M,D,N)
K(M,D,N) :=∑A⊆E
xr′(E)−r′(A)y|A|−r(A)aρ(A)−r′(A)br(A)−ρ(A),
where ρ = 12 (rmax + rmin).
I Theorems:I Contains Bollobás-Riordan-Krushkal polynomial
K(G;x,y,a,b) = bγ(G)K((M,D,N);x,y,a2,b−2)
I K(M,D,N) has a 6 term deletion-contraction relation.I duality formula, convolution formula, universality,...
I ∆-perspectives correct setting for topological Tuttepolynomials.
I Results that should hold for BRK-polynomial but donot, hold for the matroid version of the polynomial.
13
Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
11 Graphicalanalogues
Unifying TopologicalTutte polynomials
The graphical analogue
I Cellularly embedded graphs 6↔ ∆-perspectives.
I Pseudo-surface = surface withpinch points.
I Graph in pseudo surface - notnecessarily cell. embedded.
I Deletion and contraction defined in natural way:delete contract
∆-persps. ↔ graphs in pseudo-surfaces
I 7→ (B(G∗),D(G),C(G)) =: P(G)
I P(G)/e = P(G/e), P(G)\e = P(G\e), (P(G))∗ = P(G∗)
I Bollobás-Riordan-Krushkal polynomial is not apolynomial of cellularly embedded graphs.
I It is a polynomial of graphs in pseudo-surfaces.
13
Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
12 Graphicalanalogues
Unifying TopologicalTutte polynomials
The graphical analogue of subobjects
I Natural sub-objects of (M,D,N).I (M,D,N) ↔ graphsI (M,D,N) ↔ cell. embed. in surfacesI (M,D,N) ↔ cell. embed. in pseudo-surfacesI (M,D,N) ↔ non-cell. embed. in surfacesI (M,D,N) ↔ non-cell. embed. in pseudo-surfaces
I Concepts of minors, duals, etc. are compatible.
13
Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
13 Unifying TopologicalTutte polynomials
Three Topological Tutte polynomials
I Various candidates for the topological Tuttepolynomial in literature:
I M. Las Vergnas’ (1978), L(G;x,y, z)I B. Bollobás and O. Riordan’s (2001/2), R(G;x,y, z)I V. Kruskal’s (2011), K(G;x,y,a,b)
I Each corresponds to subobject
I =⇒ each polynomial is a topological Tuttepolynomial but for a different notion of embeddedgraph.
I Challenge: use this to find new combinatorialinterpretations!
13
Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
13 Unifying TopologicalTutte polynomials
Thank You!
13
Tutte polynomial
Topologicalextensions
A matroidal setting
Matroidpolynomials
Graphicalanalogues
13 Unifying TopologicalTutte polynomials
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