- Print publication year: 2015
- Online publication date: July 2015

- Publisher: Cambridge University Press
- https://doi.org/10.1017/CBO9781316106853.008
- pp 221-260

Summary

Abstract

Graph minor theory of Robertson and Seymour is a far reaching generalization of the classical Kuratowski–Wagner theorem, which characterizes planar graphs in terms of forbidden minors. We survey new structural tools and results in the theory, concentrating on the structure of large t-connected graphs, which do not contain the complete graph Kt as a minor.

1 Introduction

Graphs in this paper are finite and simple, unless specified otherwise. A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Numerous theorems in structural graph theory describe classes of graphs which do not contain a fixed graph or a collection of graphs as a minor. A classical example of such a description is the Kuratowski–Wagner theorem [92,93].

Theorem 1.1A graph is planar if and only if it does not contain K5 or K3,3 as a minor.

(We will say that G contains H as a minor, if H is isomorphic to a minor of G, and we will use the notation H ≤ G to denote this. The notation is justified as the minor containment is, indeed, a partial order. We say that G is H-minor free if G does not contain H as a minor.)

Clearly a graph is a forest if and only if it does not contain K3 as a minor. In [16] Dirac proved that a graph does not contain K4 as a minor if and only if it is series-parallel. In [93] Wagner characterizes graphs which do not contain K5 as a minor, as follows.

Theorem 1.2A graph does not contain K5 as a minor if and only if it can be obtained by 0-, 1 and 2 and 3-clique sum operations from planar graphs and V8. (The graph V8 is shown on Figure 1.)

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Surveys in Combinatorics 2015

[1] A separator theorem for nonplanar graphs, J. Amer. Math. Soc. 3 (1990), no. 4, 801–808. , , and ,

[2] Every planar map is four colorable. I. Discharging, Illinois J. Math. 21 (1977), no. 3, 429–490. and ,

[3] Every planar map is four colorable. II. Reducibility, Illinois J. Math. 21 (1977), no. 3, 491–567. , , and ,

[4] Riemann–Roch and Abel–Jacobi theory on a finite graph, Advances in Mathematics 215 (2007), no. 2, 766–788. and ,

[5] Approximating treewidth, pathwidth, frontsize, and shortest elimination tree, J. Algorithms 18 (1995), no. 2, 238–255. , , , and ,

[6] Characterizing 2-crossing-critical graphs, 2013. arXiv:1312.3712. , , , and ,

[7] Polynomial bounds for the grid-minor theorem, 2013. arXiv:1305.6577. and ,

[8] The edge-density for K2,t minors, J. Combin. Theory Ser. B 101 (2011), no. 1, 18–46. , , and ,

[9] On the maximal number of independent circuits in a graph, Acta Math. Acad. Sci. Hungar. 14 (1963), 423–439. MR0200185 (34 #84) and ,

[10] Excluding any graph as a minor allows a low tree-width 2-coloring, J. Combin. Theory Ser. B 91 (2004), no. 1, 25–41. , , , , , , and ,

[11] Graph theory, Fourth Edition, Graduate Texts in Mathematics, vol. 173, Springer, Heidelberg, 2010. ,

[12] Highly connected sets and the excluded grid theorem, J. Combin. Theory Ser. B 75 (1999), no. 1, 61–73. , , , and ,

[13] On the excluded minor structure theorem for graphs of large tree-width, J. Combin. Theory Ser. B 102 (2012), no. 6, 1189–1210. , , , and ,

[14] The Erdős-Pósa property for clique minors in highly connected graphs, J. Combin. Theory Ser. B 102 (2012), no. 2, 454–469. , , and ,

[15] Large nonplanar graphs and an application to crossing-critical graphs, J. Combin. Theory Ser. B 101 (2011), no. 2, 111–121. , , , and ,

[16] A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952), 85–92. ,

[17] Homomorphism theorems for graphs, Math. Ann. 153 (1964), 69–80. ,

[18] Layered separators in minorclosed families with applications, 2013. arXiv:1306.1595. , , and ,

[19] A stronger structure theorem for excluded topological minors, 2012. arXiv:1209.0129. ,

[20] Sublinear separators, fragility and subexponential expansion, 2014. arXiv:1404.7219. ,

[21] 2014. arXiv:1401.1399. and , List-coloring apex-minor-free graphs,

[22] Small graph classes and bounded expansion, J. Combin. Theory Ser. B 100 (2010), no. 2, 171–175. and ,

[23] Treewidth of graphs with balanced separators. manuscript. and ,

[24] Diameter and treewidth in minor-closed graph families, Algorithmica 27 (2000), no. 3-4, 275–291. Treewidth. ,

[25] Densities of minor-closed graph families, Electron. J. Combin. 17 (2010), no. 1, Research Paper 136, 21. ,

[26] Constructing dense graphs with sublinear Hadwiger number. J. Combin. Theory Ser. B, to appear. ,

[27] The highly connected matroids in minor-closed classes, 2013. arXiv:1312.5012. , , and ,

[28] Structure theorem and isomorphism test for graphs with excluded topological subgraphs, Proceedings of the fortyfourth annual acm symposium on theory of computing, 2012, pp. 173–192. and ,

[29] Über eine Klassifikation der Streckenkomplexe, Vierteljschr. Naturforsch. Ges. Zürich 88 (1943), 133–142. ,

[30] S-functions for graphs, J. Geometry 8 (1976), no. 1-2, 171–186. ,

[31] Parameters tied to treewidth, 2013. arXiv:1312.3401. and ,

[32] Cycles of given size in a dense graph, 2015. arXiv:1502.03549. and ,

[33] Tree-width and dimension, 2013. arXiv:1301.5271. , , , , , and ,

[34] Contractions to K8, J. Graph Theory 18 (1994), no. 5, 431–448. ,

[35] Densities of minor-closed graph classes are rational. in preparation. and ,

[36] Linear min-max relation between the treewidth of h-minor-free graphs and its largest grid, Lipicsleibniz international proceedings in informatics, 2012. and ,

[37] Some recent progress and applications in graph minor theory, Graphs Combin. 23 (2007), no. 1, 1–46. and ,

[38] K6 minors in 6-connected graphs of bounded tree-width. submitted. , , , and ,

[39] K6 minors in large 6-connected graphs. submitted. , , , and ,

[40] The minimum Hadwiger number for graphs with a given mean degree of vertices, Metody Diskret. Analiz. 38 (1982), 37–58. ,

[41] Lower bound of the Hadwiger number of graphs by their average degree, Combinatorica 4 (1984), no. 4, 307–316. ,

[42] Dense graphs have K3,t minors, Discrete Math. 310 (2010), no. 20, 2637–2654. and ,

[43] Treewidth and planar minors (2012). manuscript. and ,

[44] A separator theorem for planar graphs, SIAM J. Appl. Math. 36 (1979), no. 2, 177–189. and ,

[45] Well-quasi-ordering graphs by the topological minor relation: Robertson's conjecture. in preparation. and ,

[46] Graph minor theory, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 1, 75–86 (electronic). ,

[47] Homomorphiesätze für Graphen, Math. Ann. 178 (1968), 154–168. ,

[48] An excluded minor theorem for the octahedron, J. Graph Theory 31 (1999), no. 2, 95–100. ,

[49] A characterization of graphs with no cube minor, J. Combin. Theory Ser. B 80 (2000), no. 2, 179–201. ,

[50] Random graphs from planar and other addable classes, Topics in discrete mathematics, 2006, pp. 231–246. , , and ,

[51] Grad and classes with bounded expansion. I. Decompositions, European J. Combin. 29 (2008), no. 3, 760–776. and ,

[52] Characterisations and examples of graph classes with bounded expansion, European J. Combin. 33 (2012), no. 3, 350–373. , , and ,

[53] Cultivating a vortex. in preparation. and ,

[54] Linear decompositions of large graphs. in preparation. and ,

[55] Minors and linkages in large graphs. in preparation. and ,

[56] Non-planar extensions of subdivisions of planar graphs, 2014. arXiv:1402.1999. and ,

[57] Proper minorclosed families are small, J. Combin. Theory Ser. B 96 (2006), no. 5, 754–757. , , , and ,

[58] Typical subgraphs of 3- and 4-connected graphs, J. Combin. Theory Ser. B 57 (1993), no. 2, 239–257. , , and ,

[59] Forcing a sparse minor, 2014. arXiv:1402.0272. and ,

[60] The fourcolour theorem, J. Combin. Theory Ser. B 70 (1997), no. 1, 2–44. , , , and ,

[61] Graph minors. XVIII. Treedecompositions and well-quasi-ordering, J. Combin. Theory Ser. B 89 (2003), no. 1, 77–108. and ,

[62] Graph minors. XXI. Graphs with unique linkages, J. Combin. Theory Ser. B 99 (2009), no. 3, 583–616. and ,

[63] Graph minors XXIII. Nash-Williams' immersion conjecture, J. Combin. Theory Ser. B 100 (2010), no. 2, 181–205. and ,

[64] Graph minors. XXII. Irrelevant vertices in linkage problems, J. Combin. Theory Ser. B 102 (2012), no. 2, 530–563. and ,

[65] Quickly excluding a planar graph, J. Combin. Theory Ser. B 62 (1994), no. 2, 323–348. , , and ,

[66] Graph minors. I. Excluding a forest, J. Combin. Theory Ser. B 35 (1983), no. 1, 39–61. and ,

[67] Graph minors. III. Planar treewidth, J. Combin. Theory Ser. B 36 (1984), no. 1, 49–64. and ,

[68] Graph minors. II. Algorithmic aspects of tree-width, J. Algorithms 7 (1986), no. 3, 309–322. and ,

[69] Graph minors. V. Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986), no. 1, 92–114. and ,

[70] Graph minors. VI. Disjoint paths across a disc, J. Combin. Theory Ser. B 41 (1986), no. 1, 115–138. and ,

[71] Graph minors. VII. Disjoint paths on a surface, J. Combin. Theory Ser. B 45 (1988), no. 2, 212–254. and ,

[72] Graph minors. IV. Tree-width and well-quasi-ordering, J. Combin. Theory Ser. B 48 (1990), no. 2, 227–254. and ,

[73] Graph minors. IX. Disjoint crossed paths, J. Combin. Theory Ser. B 49 (1990), no. 1, 40–77. and ,

[74] Graph minors. VIII. A Kuratowski theorem for general surfaces, J. Combin. Theory Ser. B 48 (1990), no. 2, 255–288. and ,

[75] Graph minors. X. Obstructions to tree-decomposition, J. Combin. Theory Ser. B 52 (1991), no. 2, 153–190. and ,

[76] Graph minors. XI. Circuits on a surface, J. Combin. Theory Ser. B 60 (1994), no. 1, 72–106. and ,

[77] Graph minors. XII. Distance on a surface, J. Combin. Theory Ser. B 64 (1995), no. 2, 240–272. and ,

[78] Graph minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B 63 (1995), no. 1, 65–110. and ,

[79] Graph minors. XIV. Extending an embedding, J. Combin. Theory Ser. B 65 (1995), no. 1, 23–50. and ,

[80] Graph minors. XV. Giant steps, J. Combin. Theory Ser. B 68 (1996), no. 1, 112–148. and ,

[81] Graph minors. XVII. Taming a vortex, J. Combin. Theory Ser. B 77 (1999), no. 1, 162–210. and ,

[82] Graph minors. XVI. Excluding a non-planar graph, J. Combin. Theory Ser. B 89 (2003), no. 1, 43–76. and ,

[83] Graph minors. XIX. Well-quasiordering on a surface, J. Combin. Theory Ser. B 90 (2004), no. 2, 325–385. and ,

[84] Graph minors. XX. Wagner's conjecture, J. Combin. Theory Ser. B 92 (2004), no. 2, 325–357. and ,

[85] Hadwiger's conjecture for K6-free graphs, Combinatorica 13 (1993), no. 3, 279–361. , , and ,

[86] Graph searching and a min-max theorem for tree-width, J. Combin. Theory Ser. B 58 (1993), no. 1, 22–33. and ,

[87] The extremal function for K9 minors, J. Combin. Theory Ser. B 96 (2006), no. 2, 240–252. and ,

[88] Recent excluded minor theorems for graphs, Surveys in combinatorics, 1999 (Canterbury), 1999, pp. 201–222. ,

[89] An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 2, 261–265. ,

[90] The extremal function for complete minors, J. Combin. Theory Ser. B 81 (2001), no. 2, 318–338. ,

[91] A lower bound on the gonality of finite graphs, 2014. arXiv:1407.7055. and ,

[92] Sur le probléme des courbes gauches en topologie, Fundamenta Mathematicae 15 (1930), 271–283. ,

[93] Über eine Eigenschaft der ebenen Komplexe, Mathematische Annalen 114 (1937), 570–590. ,

[94] Beweis einer Abschwächung der Hadwiger-Vermutung, Math. Ann. 153 (1964), 139–141. ,