Maastricht talk - SymOmega

PERSONAL INTRODUCTION COMPUTATION

TOWARDS A CLASSIFICATION OF THE BINARY

MATROIDS WITH NO K5 MINOR

Gordon Royle

Department of Mathematics & StatisticsUniversity of Western Australia

Second Workshop on Graphs and MatroidsMaastricht, August 2010

GORDON ROYLE MATROIDS WITH NO K5 -MINOR


AUSTRALIA



AUSTRALIA



PERTH



UNIVERSITY OF WESTERN AUSTRALIA



WAGNER’S THEOREM

The foundational result of structural graph theory is thefollowing famous theorem:

THEOREM (KURATOWSKI, WAGNER)

A graph is planar if and only if it does not have the completegraph K5 or the complete bipartite graph K3,3 as a minor.

Any minor-closed class G of graphs can be described by givingits excluded minors, i.e. the minor-minimal graphs not in G.

Wagner conjectured, and Robertson & Seymour eventuallyproved, that such a list is necessarily finite.



TWO COMPLEMENTARY RESEARCH THEMES

Let G = EX (G1,G2, . . . ,Gn) be the class of graphs with nominor in {G1,G2, . . . ,Gn}.

Then we can consider two complementary research “themes”:

Find the excluded minors for some (natural) minor-closedfamily of graphs.In other words: Given G, determine {G1,G2, . . . ,Gn}.Find a structural description of the class of graphs avoidinga particular collection of minors.In other words: Given {G1,G2, . . . ,Gn}, describe G.



EXACT STRUCTURAL DESCRIPTION

Structural descriptions themselves can be qualitative or exact.Theformer determine “approximate” structure of many classes

Prominent examples in the work of Roberston, Seymour,Chudnovsky, Thomas etc.Often determines existence of polytime algorithms etc, butwith Ramsey-number type constants floating around.

The latter fill in the details for a single class

THEOREM (WAGNER)

A graph has no K5-minor if and only if it can be constructedfrom planar graphs and subgraphs of V8 by means of 0-, 1-, 2-and 3-sums.



BINARY MATROIDS

All the concepts extend essentially unchanged from graphs tobinary matroids:

Deletion, contraction, minors extend naturallyAnalogue of Wagner’s conjecture holdsQualitative structure theorems have been found by GGW

James Oxley produced various exact structural theorems in the1980s and 1990s, such as a description of the binary matroidswith no 4-wheel minor.The starting point was an old paper by Joe Kung “Excluding thecycle geometries of the Kuratowski graphs from binarygeometries” where he asked questions about growth rate,critical exponent etc.



INGREDIENTS OF A STRUCTURAL DESCRIPTION

A useful structural description of a class of matroids has thefollowing general form:

Lists of basic matroids in the class.Rules for gluing together the basic matroids to formarbitrary members of the class.

Ideally a structural description will yield a recognition algorithmwhereby an arbitrary matroid can be recursively decomposeduntil each piece can be recognised as a basic.



BINARY MATROIDS

We will view a binary matroid just as a multiset of vectors in abinary vector space GF(2)d.

0 0 0 1 1 1 10 1 1 0 0 1 11 0 1 0 1 0 1

The Fano plane F7.

(0, 0, 1)

(0, 1, 0) (1, 0, 0)

(0, 1, 1) (1, 0, 1)

(1, 1, 0)

We use columns of a matrix (of any rank) or (loose) geometricpictures to describe binary matroids.



GRAPHS AND GRAFTS

Given a graph G, the columns of the vertex-edge incidencematrix form the graphic matroid M(G).

1 2

4 3 12 13 14 23 24 34

1 1 1 1 0 0 02 1 0 0 1 1 03 0 1 0 1 0 14 0 0 1 0 1 1

The graph K4 and the binary matroid M(K4).



GRAPHS AND GRAFTS

Given a graph G, the columns of the vertex-edge incidencematrix form the graphic matroid M(G).

1 2

4 3 12 13 14 23 24 34

1 1 1 1 0 0 0 12 1 0 0 1 1 0 13 0 1 0 1 0 1 04 0 0 1 0 1 1 1

A graft is an extension of a graphic matroid.



GRAPHS AND MATROIDS

A graph theorist and matroid theorist draw K4 quite differently:

1 2

4 3

12

23

13

2414

34

To avoid confusion, we use the convention that graphs aregreen and matroids are magenta.



OUR GOAL

Our 1 major goal is to extend Wagner’s result to the class ofbinary matroids.

Obtain a structural description of the class ofbinary matroids with no M(K5) minor.

Why this particular family?

K5 and K3,3 are fundamental graphs/matroids.Various authors have studied classes defined by excludingthese in combination with other important matroids.A natural counterpart to our description of EX (M(K3,3))

1“We” = {Dillon, Geoff, Gordon}GORDON ROYLE MATROIDS WITH NO K5 -MINOR


OUR GOAL

Our 1 major goal is to extend Wagner’s result to the class ofbinary matroids.

Obtain a structural description of the class ofbinary matroids with no M(K5) minor.

Why this particular family?

K5 and K3,3 are fundamental graphs/matroids.Various authors have studied classes defined by excludingthese in combination with other important matroids.A natural counterpart to our description of EX (M(K3,3))

1“We” = {Dillon, Geoff, Gordon}GORDON ROYLE MATROIDS WITH NO K5 -MINOR


BINARY MATROIDS WITH NO M(K3,3) MINOR

THEOREM (MAYHEW, ROYLE, WHITTLE)

An internally 4-connected binary matroid in EX (M(K3,3)) isCographic, orA triadic Möbius matroid or a triangular Möbius matroid, orOne of 18 sporadic matroids.

The triangular Möbius matroid is a graft of the join K2 + Pn.



PLAN OF ATTACK

Identify the right theorem by computer-aided investigation:

Create database of low rank binary matroids in EX (M(K5))

Identify internally 4-connected and non-cographic onesFind patterns suggestive of infinite familiesIdentify sporadic “low level junk”Desperately hope low level junk peters out at tractable rank

Prove the theorem:

Find and use a suitable “chain theorem” for internally4-connected binary matroids



DATABASE CREATION

Let Γ be the bipartite point-hyperplane graph of PG(d, 2). Then:

A simple binary matroid of rank at most d + 1 with kelements is a k-set of (point-type) vertices in Γ

Two binary matroids M1 and M2 are isomorphic if and onlyif there is an automorphism of Γ mapping M1 to M2

Let Lk contain one representative of each orbit of Aut(Γ) onk-sets. An orderly algorithm can be devised that computes

Lk 7→ Lk+1.

If run to completion from L1, then — in principle — all simplebinary matroids of rank up to d + 1 are constructed once each.



MODIFIED ALGORITHM

This can be easily be modified to form lists

L′1,L′2, . . . ,L′k, . . .

containing only those matroids with no M(K5) minor.

Run as usual up to L10

Form L′10 = L10\M(K5)

At each step L′k 7→ L′k+1,Run usual algorithm on L′

k, thenTest each newly created matroid to ensure that all its singleelement minors lie in L′

` for some ` < k otherwise reject.

Straight to K5



THE 3-PRISM

To warm up, we start by excluding the 3-prism = M(K5\e)∗



JUST THE INTERNALLY 4-CONNECTED ONES

1

1 1 2 2 2 1 1 1

1 2 3 4 4 3 3 2 1

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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Size

Rank up to 3




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1 1 2 2 2 1 1 1

1 2 3 4 4 3 3 2 1

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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Rank up to 4




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1 1 2 2 2 1 1 1

1 2 3 4 4 3 3 2 1

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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8

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Rank up to 5




1

1 1 2 2 2 1 1 1

1 2 3 4 4 3 3 2 1

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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8

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Rank up to 6




1

1 1 2 2 2 1 1 1

1 2 3 4 4 3 3 2 1

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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Rank up to 7




1

1 1 2 2 2 1 1 1

1 2 3 4 4 3 3 2 1

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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Rank up to 8



NOW DONE!

The “chain theorem” for internally 4-connected binary matroidsby Carolyn Chun, Dillon Mayhew and James Oxley can beparaphrased:

THEOREM

With a small number of (specified) exceptions an internally4-connected binary matroid M contains an internally4-connected binary minor N such that |E(M)| − |E(N)| ≤ 3.

Hence 2 there are no more internally 4-connected ones, andwith a little bit more work a full structure theorem can bededuced.

2after we’ve properly checked the exceptionsGORDON ROYLE MATROIDS WITH NO K5 -MINOR


Size r = 4 r = 5 r = 6 r = 7 r = 810 3 46 105 80 3211 2 57 273 312 15112 1 62 644 1285 82113 39 1174 5110 509814 15 1390 14991 3221515 6 1161 29677 13972716 1 718 41927 41109617 331 44322 87370118 117 36059 141010719 33 22732 177905220 6 11144 178024721 1 4226 142487522 1214 91597023 260 47312824 40 19535225 4 6374726 1607927 305928 42029 4030 3



INTERNALLY 4-CONNECTED, NOT COGRAPHIC

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7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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THE MÖBIUS MATROIDS AGAIN

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7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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The triangular Möbius matroids and the triadic Möbius matroidsreappear here — with the triangular Möbius matroids havingunique maximum size by rank 7.



SOME FINITE GEOMETRY

The projective plane PG(2, 4) has points {0, 1, . . . , 20} and linesthe 21 cyclic permutations of {0, 1, 6, 8, 18}.

A hyperoval H in PG(2, 4) is a set of 6 points such that each linemeets H in 0 or 2 points.



SOME FINITE GEOMETRY

The projective plane PG(2, 4) has points {0, 1, . . . , 20} and linesthe 21 cyclic permutations of {0, 1, 6, 8, 18}.

A hyperoval H in PG(2, 4) is a set of 6 points such that each linemeets H in 0 or 2 points.



Deleting the six points of H and the six lines with no points in Hleaves the generalized quadrangle W(2).



Deleting the six points of H and the six lines with no points in Hleaves the generalized quadrangle W(2).



A FINITE FAMILY

The 21× 21 line-point incidence matrix of PG(2, 4) determines abinary matroid of rank 10; its dual is internally 4-connected, notcographic and has no M(K5) minor — call this PG(2, 4)∗

The same is true for any matroid obtained from PG(2, 4)∗ bycontracting any subset of the points of H giving us a family ofseven matroids with smallest of rank 5, size 15 and largest ofrank 11, size 21.

Fortunately PG(2, 4)∗ itself is a splitter for EX (M(K5)) and sothe family stops there.



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AN INFINITE FAMILY

This is just a ∆− Y operation on the triangle joining the threeright-most graft elements of the triangular Möbius matroid onerank higher



AN INFINITE FAMILY




AN INFINITE FAMILY




AN INFINITE FAMILY




AN INFINITE FAMILY




THE STORY SO FAR

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WHAT NEXT?

Something needs to be changed to make progress:

Try a richer base class than cographic?Perhaps rank-one perturbations of graphic, cographic andtheir duals would encompass most of them?Try different connectivity concepts to alter the “atoms” inany decomposition?Focus on narrower subclass first?

Attend European conference packed with brilliant matroidtheorists and solicit suggestions!

Thank you for your attention



WHAT NEXT?


Try a richer base class than cographic?Perhaps rank-one perturbations of graphic, cographic andtheir duals would encompass most of them?Try different connectivity concepts to alter the “atoms” inany decomposition?Focus on narrower subclass first?Attend European conference packed with brilliant matroidtheorists and solicit suggestions!




WHAT NEXT?


Try a richer base class than cographic?Perhaps rank-one perturbations of graphic, cographic andtheir duals would encompass most of them?Try different connectivity concepts to alter the “atoms” inany decomposition?Focus on narrower subclass first?Attend European conference packed with brilliant matroidtheorists and solicit suggestions!



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