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8 - Structure in minor-closed classes of matroids

Published online by Cambridge University Press:  05 July 2013

By
Jim Geelen ,
Bert Gerards  and
Geoff Whittle
Edited by
Simon R. Blackburn ,
Stefanie Gerke  and
Mark Wildon
Simon R. Blackburn
Affiliation:
Royal Holloway, University of London
Stefanie Gerke
Affiliation:
Royal Holloway, University of London
Mark Wildon
Affiliation:
Royal Holloway, University of London
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Summary

Abstract

This paper gives an informal introduction to structure theory for minor-closed classes of matroids representable over a fixed finite field. The early sections describe some historical results that give evidence that well-defined structure exists for members of such classes. In later sections we describe the fundamental classes and other features that necessarily appear in structure theory for minorclosed classes of matroids. We conclude with an informal statement of the structure theorem itself. This theorem generalises the Graph Minors Structure Theorem of Robertson and Seymour.

Introduction

For the last thirteen years we have been involved in a collaborative project to generalise the results of the Graph Minors Project of Robertson and Seymour to matroids representable over finite fields. The banner theorems of the Graph Minors Project are that graphs are well-quasi-ordered under the minor order [34] (that is, in any infinite set of graphs there is one that is isomorphic to a minor of another) and that for each minor-closed class of graphs there is a polynomial-time algorithm for recognising membership of the class [32]. We are well on track to extend these theorems to the class of F-representable matroids for any finite field F.

It is important to point out here that day-to-day work along this track does not concern well-quasi-ordering or minor testing. The actual task and true challenge is to gain insight into the structure of members of proper minor-closed classes of graphs or matroids. The well-quasi-ordering and minor-testing results are consequences – not necessarily easy ones – of the structure that is uncovered. Ironically, while one may begin studying structure with the purpose of obtaining marketable results, in the end it is probably the structural theorems themselves that are the most satisfying aspect of a project like this. To acquire that structural insight is the bulk of the work and the theorems that in the end describe the entire structure are the main deliveries of a project like this.

Type
Chapter
Information
Surveys in Combinatorics 2013 , pp. 327 - 362
Publisher: Cambridge University Press
Print publication year: 2013

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References

[1] B., Courcelle, The monadic second order logic of graphs. I. Recognizable sets of finite graphs, Information and Computation 85 (1990), 12–75.Google Scholar
[2] R., Diestel, T. R., Jensen, K. Yu., Gorbanov and C., Thomassen, Highly connected sets and the excluded grid theorem, J. Combin. Theory Ser. B 75 (1999), 61–73.Google Scholar
[3] T. A., Dowling, A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B 14 (1973), 61–83.Google Scholar
[4] J., Geelen, Some open problems on excluding a uniform minor, Adv. in Appl. Math. 41 (2008), 628–637.Google Scholar
[5] J., Geelen and B., Gerards, Excluding a group labelled graph, J. Combin. Theory Ser. B 99 (2009), 247–253.Google Scholar
[6] J., Geelen and G., Whittle, Branch width and Rota's Conjecture, J. Combin. Theory Ser. B 86 (2002), 315–330.Google Scholar
[7] J., Geelen, B., Gerards and G., Whittle, Branch-width and well-quasiordering in matroids and graphs, J. Combin. Theory Ser. B 84 (2002), 270–290.Google Scholar
[8] J., Geelen, B., Gerards and G., Whittle, On Rota's Conjecture and excluded minors containing large projective geometries, J. Combin. Theory Ser. B 96 (2006), 405–425.Google Scholar
[9] J., Geelen, B., Gerards and G., Whittle, Excluding a planar graph from GF (q)-representable matroids, J. Combin. Theory Ser. B 97 (2007), 971–998.Google Scholar
[10] J., Geelen, B., Gerards and G., Whittle, Tangles tree-decompositions and grids in matroids, J. Combin. Theory Ser. B 99 (2009), 657–667.Google Scholar
[11] J., Geelen and K., Kabell, Projective geometries in dense matroids, J. Combin. Theory Ser. B 99 (2009), 1–8.Google Scholar
[12] J., Geelen, J., Kung and G., Whittle, Growth rates of minor-closed classes of matroids, J. Combin. Theory Ser. B 99 (2009), 420–427.Google Scholar
[13] J., Geelen and G., Whittle, Cliques in dense GF (q)-representable matroids, J. Combin. Theory Ser. B 87 (2003), 264–269.Google Scholar
[14] J., Geelen and G., Whittle, Inequivalent representations of matroids over prime fields, submitted.
[15] T., Gowers, The two cultures of mathematics, in Mathematics, Frontiers and Perspectives (ed. V., Arnold, M., Atiyah, P., Lax, B., Mazur), American Mathematical Society, Rhode Island (2000).
[16] T., Gowers, Rough structure and classification, in Geom. Funct. Anal., Special Volume, Part I (2000), 79–117.
[17] I., Heller, On linear systems with integral valued solutions, Pacific J. Math. 7 (1957), 1351–1364.Google Scholar
[18] I. V., Hicks and N. B., McMurray, The branchwidth of graphs and their cycle matroids, J. Combin. Theory Ser. B 97 (2007), 681–692.Google Scholar
[19] P., Hlinený, Branch-width, parse trees and monadic second-order logic for matroids, J. Combin. Theory Ser. B 96 (2006), 325–351.Google Scholar
[20] P., Hlinený, Some hard problems on matroid spikes, Theory of Computing Systems 41 (2007), 551–562.Google Scholar
[21] P., Hlinený and G., Whittle, Matroid tree width, European J. Combin. 27 (2006), 1117–1128.Google Scholar
[22] T., Huynh, 2009. The linkage problem for group-labelled graphs, PhD Thesis, University of Waterloo,
[23] J., Kahn and J. P. S., Kung, Varieties of combinatorial geometries, Trans. Amer. Math. Soc. 271 (1982), 485–499.Google Scholar
[24] J. P. S., Kung, Extremal matroid theory, Contemporary Mathematics 147 (1993), 21–61.Google Scholar
[25] W., Mader, Homomorphieeigenschaften und mittlere Kantendichte von Graphen, Math. An. 174 (1967), 265–268.Google Scholar
[26] D., Mayhew, M., Newman and G., Whittle, On excluded minors for real representability, J. Combin. Theory Ser. B 99 (2009), 685–689.Google Scholar
[27] D., Mayhew, M., Newman, D., Welsh and G., Whittle, On the asymptotic proportion of connected matroids, European Journal of Combinatorics 32 (2011), 882–890.Google Scholar
[28] J. G., Oxley, Matroid Theory, Oxford University Press, Oxford (2011).
[29] J. G., Oxley, What is a matroid?, Cubo 5 (2003), 179–218.Google Scholar
[30] N., Robertson and P. D., Seymour, Graph Minors. V. Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986), 92–114.Google Scholar
[31] N., Robertson and P. D., Seymour, Graph Minors. X. Obstructions to tree decomposition, J. Combin. Theory Ser. B 59 (1991), 153–190.Google Scholar
[32] N., Robertson and P. D., Seymour, Graph Minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B 63 (1995), 65–110.Google Scholar
[33] N., Robertson and P. D., Seymour, Graph Minors. XVI. Excluding a non-planar graph, J. Combin. Theory Ser. B 89 (2003), 43–76.Google Scholar
[34] N., Robertson and P. D., Seymour, Graph Minors. XX. Wagner's Conjecture, J. Combin. Theory Ser. B 92 (2004), 325–357.Google Scholar
[35] N., Robertson, P. D., Seymour and R., Thomas, Quickly excluding a planar graph, J. Combin. Theory Ser. B 62 (1994), 323–348.Google Scholar
[36] G. -C., Rota, Combinatorial theory, new and old, in Actes du Congres International des Mathematiciens (Nice, Sept. 1970) Tome 3, pp. 229–233, Gaulthier-Villars, Paris (1971).
[37] P. D., Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305–354.Google Scholar
[38] P. D., Seymour, Triples in matroid circuits, European J. Combin. 7 (1986), 177–185.Google Scholar
[39] P., Turán, On an extremal problem in graph theory, Matematikai és Fizikai Lapok 48 (1941), 436–452.Google Scholar
[40] K., Truemper, On the efficiency of representability tests for matroids, European J. Combin. 3 (1982), 275–291.Google Scholar
[41] W. T., Tutte, A homotopy theorem for matroids, I, II, Trans. Amer. Math. Soc. 88 (1958), 144–174.Google Scholar
[42] T., Zaslavsky, Biased graphs. II. The three matroids, J. Combin. Theory Ser. B 51 (1991), 46–72.Google Scholar

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