A CONSTRUCTION FOR A FAMILY OF SETS AND ITS

APPLICATION TO MATROIDS

J.E. DAWSON

Given a family A of subsets of an ordered set E, we define a

construction giving a family of sets B in i-i correspondence with A; the

same construction applied to B then gives A. For each subset X of E,

AAB C X C AUB for exactly one pair of AEA and corresponding B6B. When the

family B is the basis collection of a matroid on E, n can be described

simply in terms of the matroid structure. A polynomial is defined which,

in this latter case, is the Tutte polynomial of the matroid.

I. INTRODUCTION

Let E be a finite totally ordered set which we will write as

{l,2,...,n}, and let m denote {1,2,...,m}(so that n = E). For eCE, we

will often write e for {e}. Let A CP(E), where A is non-empty, and call its

members A-sets. We first associate a member of A with each subset of E, by

defining a map a:P(E)+A.

Definition I.I.

d(X,Y) = w((X\Y) U (Y\X)).

minimizes d(G,~(G)).

For ICE, let w(I) = ~ 2 i, and for x,yC_E, let i6I

For GCE, let ~(G) be that member of n which

It is trivial that w:P(E)+IN is an injective map, and so the above

definition defines ~(G) uniquely. Also, if X,Y~E with z = max((X\Y) U (Y\X)),

then w(X) > w(Y) if zEX\Y and w(Y) > w(X) if z6Y\X. Thus we have the

following result characterizing whether or not a particular element m is in

~(G).

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Lemma i. 2.

(a) if every A-set A such that A\m = e(G)\m (does, does not)

contain m, then m (is, is not) in c~(G), and

(b) if there are A-sets A', A" such that A'\m = A"\m = c~(G)\~n

and m@A' but m~A", then m (is, is not) in o~(G) if m (is, is not) in G.

Proof Since e(G)CA,(a) is trivial. Suppose the condition

of clause (b)holds but that mC~(G). Then,

d(G,~(G)) ~ 2 m + ~{2i: iE((~(G)) U(~(G)\G))\m}

whereas

d(G,A') ~ 21+22 +...+ 2 m-I + ~{2i: ie((~(G)) U (~(G)\G))\~}

Since 21+22 +...+ 2 m-I < 2 TM, this contradicts the definition of ~(G). A

similar proof applies if mE~(G)\ G. Thus (b) holds. []

We note that Lemma 1.2 determines ~(G)\m-i in terms only of

~(G)\m, ~m-i and {~m-i : AEA}. Thus we can use it to construct ~(G) from

G and A, by applying it successively to the cases m=n, m=n-l, ... , m=l.

Also, it could have been used as the definition of the map ~:P(E)§ To do

so we would need to see that ~(G) is indeed in A; however it is quite clear

that if A\m = ~(G)\m for some AEA, and ~(G)\m-i is determined using Lemma

1.2, then A\m-i = ~(G)\m-i for some AEA. We can describe the construction

thus: "to change G into ~(G), we work from the highest element downwards,

making changes only where necessary to get ~(G)6A".

Example Writing {1,2} as 12, let n=3 and A = {12,13,23}. Then

~-i(12) = {~,i,2,12}, ~-i(13) = {3,13} and ~-i(23) = {23,123}.

From the example we notice that each e-l(A) is of the form

{XCE: HCXCF}, an interval of the lattice of subsets of E (a 2attice interva2),

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which we write <H,F>. So, for AEA, we now determine ~-I(A), showing that it

is always of this form.

Let H = {m6A: there exists A'EA such that A'\m = A\m and m~A'},

and F = ~ {m6E\A: there exists A'EA such that A'\m = A\m and m6A'}.

Lemma 1.3. For AeA, e-l(A) = <H,F>.

Proof It is clear from Lemma 1.2(b) that ~-I(A) C <H,F>.

Suppose G~-I(A); say d(G,A) > d(G,A') for some A'eA and let

m = max((A\A') U (A'\A)). Suppose mEA'\A. Then mEEkF, but clearly meG.

If, on the other hand, m6A\A', then m~-H, and clearly m~G. Thus G ~ <H,F>,

as required. []

Since HCACF, writing H = AAB and F = AUB uniquely defines a set B

(B = Hu(~ A)), which we describe next.

Lemma 1.4. For AeA, ~-I(A) = <AA6'(A), AU6'(A)> where

6':A§ is defined by:

m66'(A) if and only if either

(a) for every A-set A' such that A'\m = ~ m, m~A', or

(b) meA, and there is A'eA such that A'\m = A\m and m~A'.D

This result gives a partitioning of P(E) into lattice intervals:

P(E) = U{~-I(A): AEA}. Let us define 6 = im(6'). Then ~IB and 6' are

mutually inverse bijections between A and 6, with the property that

6'(A)\m = 6'(A')\m if and only if A\m = A'\m.

In the example above, 8'(12) = r B'(13) = 3, 6'(23) = 123, and

B = {~,3,123}.

Theorem 1.5. Let AEA, B = 8' (A) and mEE. Then la, 2a and 3a

are equivalent, as are ib, 2b and 3b.

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(la,lb)

(2a,2b)

(3a,3b)

(m6B\ A, mEA\ B)

All A-sets A' such that A'\m = Aim (do not, do)

contain m.

All B-sets B" such that B"\m = B\m (do, do not)

contain m.

Proof la ~ 2a and ib ~ 2b follow from the definition of B' (also

from Lemma 1.2.) 2a = 3a: assume 2a holds, and suppose B"\m = B\m and

A" = ~(B"). Thus A"\m = A\m. Now condition 2a remains true for ~' as it is

for A, and so, by the definition of B', B" = B'(A") contains m. Thus 3a

follows. 2b = 3b is shown similarly.

3a = 2a: assume 3a. Thus meB, 3b does not hold, and

so 2b does not hold. Let ~'\m = A\m with m~A". Hence B'(N')\m = B\m, and by

the assumption 3a, mEB'(A"). Thus, by the definition of ~', and since

A"\m = A\m, 2a holds.

3b ~ 2b is shown similarly. []

There is an obvious symmetry between (2a,2b) and (3b,3a) of

Theorem 1.5. We define ~':8+P(E) in terms of B just as ~' is defined in

terms of A in Lemma 1.4; likewise we define B:P(E)+B in terms of B in a

manner analogous to Definition i.i.

Corollary 1.6. Im(~') = A, and ~' = ~IB = (~,)-i = (BIA)-I.

Proof As with ~IB and B', ~' and Blim(~') are mutually

inverse bijections. Just as Theorem 1.5, (la,lb) ~ (2a,2b), contains the

definition of ~', (la,lb) ~ (3a,3b) asserts that A = ~'(B) for A,B as in the

Theorem (equivalently, for BEB and A = ~(B)). Thus ~' = ~IB, so im(~') = A;

just as (8,)-1 = ~IB, (~,)-I = BIA, and the result follows. []

We can now refer to ~' as ~; we simply have one description of

in terms of A, and one of ~IB in terms of B. Similarly for B. The following

Corollary is useful.

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Corollary i. 7.

(i) ~ = ~ and ~2 = ~.

(ii) ~(G) = o.(H) ~> ~(G) = ~(H), and, in this case, if

G~HCFCGUH then ~(F) = ~(G).

(iii) For GC_E, ~(G)NB(G)CGC~(G)UB(G)

(iv) Let gEG. Then g~B(G) ~ gEa(G\g), and in this case

~(G) = ~(G\g).

In each of the above results we may interchange ~ and B.

Proof (i) Let AEA and B = B(A). It was shown earlier that

-i (A) = <ANB,AUB>, and similarly B-I(B) = <AAB,AUB>. As ~I B is a bijection,

(eB)-I(A) = B-I(B), and the first result follows. Clearly a is idempotent.

(ii) The equivalence holds as ~B = ~ and similarly

B~ = 8. Also ~-I(~(G)) = <~(G) n B(G), ~(G) u ~(G)>, which is a lattice

interval; since it contains G and H, it also contains F.

(iii) follows as GE ~-I(~(G)) = <~(G) N B(G), ~(G) U B(G)>.

(iv) Let g~ B(G). Then G\g E ~-l(e(G)) and so

a(G\g) = a(G). Now G C a(G) U B(G), so g@ ~(G) = ~(G\g) as required. The

converse proof is similar. []

Let us call A and B, as above, a complementary pair of subsets of

P(E), or a complementary pair on E. We can form from A and B some other

complementary pairs.

Theorem 1.8. If A and B are a complementary pair of subsets of

P(E), then each of the following also forms a complementary pair:

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(i) on E: A* = {E\A: Ae A}, where ~*(E\G) = E\~(G), with

8" and 8" defined similarly,

(ii) on E\m for mEE: A' = {Akm:AEA} where ~'(~m) = ~(G)\m,

with ~' and 8' defined similarly.

(iii) For m6E, let A'eA' of (ii), and B' = 8'(A'). Let A"=

{AC~:AeA, A\m = A'}, and for G such that ~m = A', let ~"(GC~) = ~(G)~.

Then with ~" and 8" defined similarly, A" and 8" form a complementary

pair on EC~.

Proof (i) is easy. For (ii), note that by the comment following

Lemma 1.2, e is well-defined. (iii) is now straightforward. []

2. SOME ASSOCIATED OPERATORS

We define some further operators related simply to a and 8. Let

y(G) = ~(G) U 8(G) and 6(G) = a(G) A 8(G). Then by Corollary 1.7 (i) and

(ii) we have that a, 8, Y and 6 have the property that for

{~,n} ~ {~,8,u ~n = ~. Thus

~-I(~(G)) = 8-I(8(G)) = y-l(y(G)) = 6-I(6(G)) = <6(G), (G),.

We can also define

~(G) = B(G) U G, 0(G) = B(G) N G, ~(G) = ~(G) U G and @(G) = e(G) A G

(which are all contained in <6(G), y(G)>).

Proofs in this section are brief or absent; they use Lemma 1.4 and

Corollary 1.7 (i) to (iii), and are easily seen with the help of a Venn

diagram displaying G, ~(G) and 8(G).

Lemma 2.1 The operators ~,~,0,~ are idempotent, and

~ = ~, ~ = ~, O~ = 0 and ~0 = ~; otherwise they commute, with

~ = y, O~ = 6, ~O = 8 and ~ = ~. []

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Lemma 2.2 (i) r = ~-I(~(G)) = <9(G),~(G)> and

0-1(e(G)) = w-l(~(G)) = <0(G),~(G)>.

(ii) For g6G, g~(G) ~ g6#(~g); similarly e and ~.

Proof (i) The first equality follows as ~ = ~ and ~ = ~.

Likewise, B~ = 6 and so #(H) = ~(G) implies 6(H) = 6(G). As ~(H) = H U 6(H),

it is now easy to check that

~-I(~(G)) = {HE<d(G), y(G)>: ~6(H) = ~6(G)}

= <~(G),r

The second result is shown similarly.

(ii) This is immediate from Corollary 1.7 (iv). []

The last Lemma gives two partitions of P(E) into lattice intervals,

which are subintervals of <~(G),y(G)>, with the following property.

Lemma 2.3 ~(H) = ~(G) and e(H) = 8(G) = G = H, i.e.,

<~(G),~(G)> n <O(G), ~(G)> = {G}.

Proof G\e(G) = G\B(G) = ~(G)\6r since 6~ = B. Thus O(G)

and ~(G) determine G. []

Although ~(G) and ~(G) may vary even for G within a fixed

<~(H),y(H)>, ~(G)\~(G) may not, for ~(G)\~(G) = 6(G)\~(G); similarly

e(G)\ m(G) = ~(G)\ 6(G). Finally we pair these operators in another way.

Lemma 2.4 G6<O(H), r ~ HE<~(G),~(G)>, and in this case,

e(H) C 8(G) C G C ~(G) C ~(H) ,

~(G)\G C ~(H)\H, G\e(G) C Hke(H) ,

and also the similar statements got by interchanging G with H, ~ with ~ and

@ with 4.

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Proof Either of the first conditions gives e(G) = ~(H) etcetera.

The proof is then easy with a Venn diagram of G,H,~(G) and B(G). []

Although these results are quite simply proved with the suggested

Venn diagrams, they are of interest when B is the basis collection of a

matroid on E.

3. APPLICATION TO MATR01DS

Suppose we let B be the basis collection of a matroid E on E.

That is, B is a non-empty clutter on E (i.e. no B-set contains another) and

E is the collection of subsets of sets in B, called the independent sets,

satisfying the axiom: for FEE and BEB, there exists B'CB such that

F C B' C F U B. Let C be the collection of the minimal dependent sets,

called circuits. Let B* = {E\B: BEB}, as in Theorem 1.8. Then B* is the

basis collection of another matroid E* (the dual of E), whose circuit

collection C* we also call D, the cecircuit collection of the original

matroid E. We say that e depends on F, elF or eE[F], if eEF or there

exists C6C such that e6C C F U e. Similarly ]* and [ ]* are defined

relative to C* (=Q). (We note that the matroid property of B is preserved

also by construction (iii), but not by (ii), of Theorem 1.8.) It can be

shown that [ ] has the properties of a closure operator

(i.e. F C [F] and F C [G] ~ [F] C [G]), that for e~F, e6[F] ~ e~\F\e]*,

and that bases have the same number df elements. If G C E we define the

matroid EIG = {F ~ G: FEE} on G. For further details see [6], [4] or [3].

For such B, the maps e and B have a particularly simple description in terms

of the matroid structure.

Theorem 3.1 Let B be the collection of bases of a matroid on

E. Then, for G ~ E,

(i) ~(G) = GU{e~G: eiG\e}k{e6G: eI*E\Gke}

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(ii) B(G) ~- GU {e~G: eI*E\G\e}\{eEG: eIG\e} ,

which we can write as

B(G) = GU {rain(D): DED, DNG = r CeC, CCG}

or B(G) = (E\G)U {max(D~G) : DED, D~G r O}\{max(CkG): CEC, C\G # 0}.

Proof Note that we could equally write these results with "6G"

and "@G" replaced by "EE" since it is not possible that eIG\e and eI*E\Gke.

It is easy to check the equivalence of the three descriptions of 6(G).

Firstly we show that (i) is true for G = BEB, using the definition of ~ in

terms of g (like that of B in terms of A in Lemma 1.4). Let eEtB. If eIB\e,

then clearly every B'EB such that B'\e = B\e does not contain e, and eel(B).

However, if e+B\e, then B\~U{e}EE, and so there is a basis B' such that

B\e U{e} c B' C BU{e}, and so B'\e = B\e, with eEB'. Therefore e~(B).

Similarly we can verify (i) with respect to the elements e in B(=G).

Further, since BEE and E\B E E*, we can write e(B)\B = {eEE: eIB\e } and

B\~(B) = {eEE: e]*E\B\e}. To complete the proof, we need the next Lemma.

Lemma 3.2 For GE<BCh(B), BUs(B)>, eE[G\e] ,~ eE[B\e] and

ee[E\ G\ e]* ~> eE[E\ B\ e]*.

Proof First we show by induction on IG\BI that [G\e] C__ [B\e].

Let gEG\B (Ce(B)kB), so ge[B\g], and assume by induction thatthis result is

true for G\g (for all e). If g.<e then [G\e] = [G\gke] C [B\e]. If g>e,

then ge[B\g] C [B\e]. As [G\g\e] C [B\e], we have [G\e] C [Bke].

Now suppose eE[B\e]\ [G\e]. Then for some bEB\G\ e, some circuit

C C B\~U{e} contains e and b, whereby blB\bUe. As beB\G, bl*E\B\b;

since eEq~, this contradicts b IB\bUe. Thus the first result follows. The

second part of the Lemma is shown similarly. []

Proof of Theorem (continued). Let GC_E, and recall that

GE<BC~(B), BUs(B)> for B = 6(G). Then

145

~(G) = ~(B) = GU(~(B)\B)\ (B\~(B))

= GU{eEE:eIB\e} \ {eeE:ei*E\B\e}

= GU{eEE:eIG\e } \ {eEE:eI*E\G\e } ,

as required. For (ii), we have, for e~G,

e6B(G) ~ e~(GUe) (symmetrical with Corollary 1.7 (iv))

el*E\ (GUe)\ e

el*E\ G\ e .

We complete the proof of (ii) by a similar argument. []

Corollary 3.3 AeA if and only if ~(A) = A, i .e.,

(i) if CEC and C\{min(C)} C_ A, then C C_ A, and

(ii) if D6~ and D\{min(D)} C E\A, then D C E\A. []

The corresponding result for ~ turns out to be that B6B if and

only if B contains no circuit and E\B contains no cocireuit. It is shown

in [5] that the (unique) lexicogr~phic maximum basis B = {b I ..... b n}

is "maximum" in the stronger sense that for any other basis

F = [fl,...,fn}, bi>.f i for each i when B and F are each written with

elements in ascending order. The next result then follows easily from [5]

or [2].

Proposition 3.4 For ~ C__ E, @(G) is the maximum basis of

EIG and E\~(G) is the maximum basis of E* I(E\G). []

We now see that some of the results of [I], w are applications

to matroids of results in Sections 1 and 2. (Note that to get our results

into the form of those in [i], we need to reverse the ordering of the set.)

Thus IG\~(G) I = IG\~(G) I is the internal activity i(G) of G, and I~(G)\G I

is its external activity e(G). It follows from Proposition 3.4 that IG\B(G) I

is its dependence, d(G) = IGI - r(G) (where r(G) is the rank of G), and that

146

IB(G)\G I is r(E)-r(G), which we may call the shortage of G, s(G). Let us

retain these definitions of i,e,d and s based on ~ and 8, but drop the

assumption that B is a matroid basis collection.

Now if G C E, A = ~(G) and B = B(G), then ACB C G C At)B, and so

e(B) = d(A) = IA\BI = [A\G I + IG\BI = e(G) + d(G).

Similarly i(B) = s(A) = i(G) + s(G). Also, the intervals <A~B,AUB> each

contain one A-set and one B-set. (In the matroid case, these are the

structureless intervals of [i], Proposition 12.) Such an interval contains

Is(A) I [d(A) 1 sets G such that s(G)= i and d(G)= j, since such G is of

~ i J ~ " the form

B U {j elements of A\B} \ {i elements of B\A} .

This gives the result, shown by Crapo for the matroid case ([i], Theorem i),

that if T(B;x,y) = E{x i(B) ye(B): BEB} , which is also equal to E{x s(A) yd(A): AEA},

d(G) and R(B;x,y) = ~{x s(G) y : G C E}, then R(B;x,y) = T(B; x+l,y+l).

Proposition 3.5 R(B;x,y) = E{x i(G) ye(G): G ~ E}

Proof A similar argument shows that the right-hand side is

equal to T(B;x+l,y+l). []

It is now useful to return to the assumption that B is a matroid

basis collection, for then (see [i]) R(B;x,y) does not depend on the

ordering of E, and hence neither does T. This tells us that the number of

A-sets of given dependence and shortage is independent of the ordering of E.

The i-i correspondence between A, B, im(y) and im(B), established

in Section 2, has some interesting consequences for matroids. In particular,

Theorem 3.6 (ii) relates to the fact that matroids seem usually to have more

bases than circuits ([6], p.74).

Theorem 3.6 A matroid has at least as many bases as (i) fully

dependent flats (ii) proper upper segments of circuits (i.e.,

Clmin(C)\m, for CeC, meE).

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Proof (i) Fully dependent flats are in A.

(ii) Such a set F satisfies ~(F) D F and B(F) D F.

As FD6(F), so F = B(F) and F is in im(6), which is in i-1 correspondence with

B. []

I wish to thank Dr. C.C. Heyde for arousing my interest in

this subject.

REFERENCES

[1]

[2]

[3 ]

[4]

[5]

[6]

H.H. Crapo, The Tutte polynomial, Aeq. Math. 3 (1969), 211-229.

J.E. Dawson, Optimal matroid bases: an algorithm based on

eocircuits, Quart. J. Math. (Oxford) (2) 31 (1980), 65-69.

J.E. Dawson, A simple approach to some basic results in matroid

theory. J. Math. Anal. AppI., 75 (1980), 611-615.

R. von Randow, Introduction to the Theory of Matroids, Lecture

Notes in Economics and Mathematical Systems 109, Springer-Verlag, 1975.

D.J.A. Welsh, Kruskal's theorem for matroids, Proc. Camb. Phil. Soc.

64 (1968), 3-4.

D.J.A. Welsh, Matroid Theory, Academic Press, New York, 1976.

Division of Mathematics and Statistics, CSIRO, P.O. Box 218, LINDFIELD, N.S.W. 2070, AUSTRALIA.