The world of hereditary graph classes viewed through Truemper configurations

doi:10.1017/CBO9781139506748.008

7 - The world of hereditary graph classes viewed through Truemper configurations

Published online by Cambridge University Press: 05 July 2013

Edited by

Stefanie Gerke and

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

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of a few specific types, that we will call Truemper configurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices).

The configurations that Truemper identified in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper configurations, focusing on the algorithmic consequences, trying to understand what structurally enables efficient recognition and optimization algorithms.

Introduction

Optimization problems such as coloring a graph, or finding the size of a largest clique or stable set are NP-hard in general, but become polynomially solvable when some configurations are excluded. On the other hand they remain difficult even when seemingly quite a lot of structure is imposed on an input graph. For example, determining whether a graph is 3-colorable remains NP-complete for triangle-free graphs with maximum degree 4 [92].

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Type: Chapter
Information: Surveys in Combinatorics 2013 , pp. 265 - 326

DOI: https://doi.org/10.1017/CBO9781139506748.008 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2013

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References

[1] L., Addario-Berry, M., Chudnovsky, F., Havet, B., Reed, and P., Seymour, Bisimplicial vertices in even-hole-free graphs, Journal of Combinatorial Theory Series B 98 (2008), 1119–1164.Google Scholar

[2] P., Aboulker, P., Charbit, M., Chudnovsky, N., Trotignon, and K., Vuškovic, LexBFS, structure and algorithms, preprint (2012), submitted.

[3] P., Aboulker, M., Radovanovic, N., Trotignon, and K., Vuškovic, Graphs that do not contain a cycle with a node that has at least two neighbors on it, to appear in SIAM Journal on Discrete Mathematics.

[4] P., Aboulker, M., Radovanovic, N., Trotignon, T., Trunck, and K., Vuškovic, Linear balanceable and subcubic balanceable graphs, preprint (2012), submitted.

[5] C., Berge, Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind (Zusammenfassung), Technical report, Wiss. Z. Martin Luther Univ. Math.-Natur. Reihe (Halle-Wittenberg) (1961).

[6] C., Berge, Sur Certains Hypergraphes Généralisant les Graphes Bipartis, in Combinatorial Theory and its Applications I (eds. P., Erdös, A., Rényi and V., Sós), Colloquia Mathematica Societatis János Bolyai 4, North-holland, Amsterdam (1970), 119–133.

[7] M., Burlet and J., Fonlupt, Polynomial algorithm to recognize a Meyniel graph, Discrete Math 21 (1984), 225–252.Google Scholar

[8] K., Cameron, Antichain sequences, Order 2 (1985), 249–255.Google Scholar

[9] K., Cameron, B., Lévêque, and F., Maffray, Coloring vertices of a graph or finding a Meyniel Obstruction, Theoretical Computer Science 428 (2012), 10–17.Google Scholar

[10] P., Camion, Characterization of totally unimodular matrices, Proceedings of the American Mathematical Society 16 (1965), 1068–1073.Google Scholar

[11] H.-C., Chang and H.-I., Lu, A faster algorithm to recognize even-holefree graphs, Proceedings of 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (2012), 1286–1297.

[12] P., Charbit, M., Habib, N., Trotignon, and K., Vuškovic, Detecting 2-joins faster, to appear in Journal of Discrete Algorithms.

[13] P., Charbit, F., de Montgolfier, and M., Raffinot, Linear time split decomposition revisited, SIAM Journal on Discrete Mathematics 26 (2) (2012), 499–514.Google Scholar

[14] V., Chvátal and N., Sbihi, Bull-free Berge graphs are perfect, Graphs and Combinatorics 3 (1987), 127–139.Google Scholar

[15] V., Chvátal and N., Sbihi, Recognizing claw-free Berge graphs, Journal of Combinatorial Theory Series B 44 (1988), 154–176.Google Scholar

[16] M., Chudnovsky, Berge trigraphs and their applications, PhD thesis, Princeton University (2003).

[17] M., Chudnovsky, Berge trigraphs, Journal of Graph Theory 53 (1) (2006), 1–55.Google Scholar

[18] M., Chudnovsky, The structure of bull-free graphs I: Three-edge paths with centers and anticenters, Journal of Combinatorial Theory Series B 102 (1) (2012), 233–251.Google Scholar

[19] M., Chudnovsky, The structure of bull-free graphs II and III: A summary, Journal of Combinatorial Theory Series B 102 (1) (2012), 252–282.Google Scholar

[20] M., Chudnovsky, The structure of bull-free graphs II: elementary trigraphs, manuscript.

[21] M., Chudnovsky, The structure of bull-free graphs III: global structure, manuscript.

[22] M., Chudnovsky, G., Cornuéjols, X., Liu, P., Seymour, and K., Vuškovic, Recognizing Berge graphs, Combinatorica 25 (2) (2005), 143–186.Google Scholar

[23] M., Chudnovsky and R., Kapadia, Detecting a theta or a prism, SIAM Journal on Discrete Math 22 (2008), 1164–1186.Google Scholar

[24] M., Chudnovsky, K., Kawarabayashi, and P., Seymour, Detecting even holes, Journal of Graph Theory 48 (2005), 85–111.Google Scholar

[25] M., Chudnovsky and I., Penev, The structure of bull-free perfect graphs, to appear in Journal of Graph Theory.

[26] M., Chudnovsky, N., Robertson, P., Seymour, and R., Thomas, The strong perfect graph theorem, Annals of Mathematics 164 (1) (2006), 51–229.Google Scholar

[27] M., Chudnovsky and S., Safra, The Erdős-Hajnal conjecture for bullfree graphs, Journal of Combinatorial Theory Series B 98 (2008), 1301–1310.Google Scholar

[28] M., Chudnovsky and P., Seymour, Solution of three problems of Cornuéjols, Journal of Combinatorial Theory Series B 98 (2008), 116–135.Google Scholar

[29] M., Chudnovsky and P., Seymour, The three-in-a-tree problem, Combinatorica 30 (4) (2010), 387–417.Google Scholar

[30] M., Chudnovsky and P., Seymour, Claw-free Graphs I: Orientable prismatic graphs, Journal of Combinatorial Theory Series B 97 (2007), 867–903.Google Scholar

[31] M., Chudnovsky and P., Seymour, Claw-free Graphs II: Nonorientable prismatic graphs, Journal of Combinatorial Theory Series B 98 (2008), 249–290.Google Scholar

[32] M., Chudnovsky and P., Seymour, Claw-free Graphs III: Circular interval graphs, Journal of Combinatorial Theory Series B 98 (2008), 812–834.Google Scholar

[33] M., Chudnovsky and P., Seymour, Claw-free Graphs IV: Decomposition theorem, Journal of Combinatorial Theory Series B 98 (2008), 839–938.Google Scholar

[34] M., Chudnovsky and P., Seymour, Claw-free Graphs V: Global structure, Journal of Combinatorial Theory Series B 98 (2008), 1373–1410.Google Scholar

[35] M., Chudnovsky and P., Seymour, Claw-free Graphs VI: Colouring, Journal of Combinatorial Theory Series B 100 (2010), 560–572.Google Scholar

[36] M., Chudnovsky and P., Seymour, Claw-free Graphs VII: Quasi-line graphs, Journal of Combinatorial Theory Series B 102 (2012), 1267–1294.Google Scholar

[37] M., Chudnovsky, N., Trotignon, T., Trunck, and K., Vuškovic, Coloring perfect graphs with no balanced skew-partitions, preprint (2012), submitted.

[38] V., Chvátal, Star cutsets and perfect graphs, Journal of Combinatorial Theory Series B 39 (1985), 189–199.Google Scholar

[39] V., Chvátal and N., Sbihi, Bull-free Berge graphs are perfect, Graphs and Combinatorics 3 (1987), 127–139.Google Scholar

[40] S., Cicerone and D., Di Stefano, On the extension of bipartite graphs to parity graphs, Discrete Applied Mathematics 95 (1999), 181–195.Google Scholar

[41] M., Conforti, G., Cornuéjols, G., Gasparyan, and K., Vuškovic, Perfect graphs, partitionable graphs and cutsets, Combinatorica 22 (1) (2002), 19–33.Google Scholar

[42] M., Conforti, G., Cornuéjols, A., Kapoor, and K., Vuškovic, Universally signable graphs, Combinatorica 17 (1) (1997), 67–77.Google Scholar

[43] M., Conforti, G., Cornuéjols, A., Kapoor, and K., Vuškovic, Even and odd holes in cap-free graphs, Journal of Graph Theory 30 (1999), 289–308.Google Scholar

[44] M., Conforti, G., Cornuéjols, A., Kapoor, and K., Vuškovic, Balanced 0, ±1 matrices, Part I: Decomposition theorem, and Part II: Recognition algorithm, Journal of Combinatorial Theory Series B 81 (2001), 243–306.Google Scholar

[45] M., Conforti, G., Cornuéjols, A., Kapoor, and K., Vuškovic, Evenhole-free graphs, Part I: Decomposition theorem, Journal of Graph Theory 39 (2002), 6–49.Google Scholar

[46] M., Conforti, G., Cornuéjols, A., Kapoor, and K., Vuškovic, Even-holefree graphs, Part II: Recognition algorithm, Journal of Graph Theory 40 (2002), 238–266.Google Scholar

[47] M., Conforti, G., Cornuéjols, and M. R., Rao, Decomposition of balanced matrices, Journal of Combinatorial Theory Series B 77 (1999), 292–406.Google Scholar

[48] M., Conforti, G., Cornuéjols, and K., Vuškovic, Square-free perfect graphs, Journal of Combinatorial Theory Series B 90 (2004), 257–307.Google Scholar

[49] M., Conforti, G., Cornuéjols, and K., Vuškovic, Decomposition of oddhole-free graphs by double star cutsets and 2-joins, Discrete Applied Mathematics 141 (2004), 41–91.Google Scholar

[50] M., Conforti, G., Cornuéjols, and K., Vuškovic, Balanced matrices, Discrete Mathematics 306 (2006), 2411–2437.Google Scholar

[51] M., Conforti and A. M. H., Gerards, Stable sets and graphs with no even holes, preprint (2003), unpublished.

[52] M., Conforti, B., Gerards, and A., Kapoor, A theorem of Truemper, Combinatorica 20 (1) (2000), 15–26.Google Scholar

[53] M., Conforti and M. R., Rao, Structural properties and decomposition of linear balanced matrices, Mathematical Programming 55 (1992), 129–168.Google Scholar

[54] M., Conforti and M. R., Rao, Testing balancedness and perfection of linear matrices, Mathematical Programming 61 (1993), 1–18.Google Scholar

[55] G., Cornuéjols, Combinatorial Optimization: Packing and Covering, SIAM, Philadelphia, PA (2001). CBMS-NSF Regional Conference Series in Applied Mathematics 74.

[56] G., Cornuéjols and W. H., Cunningham, Composition for perfect graphs, Discrete Mathematics 55 (1985), 245–254.Google Scholar

[57] B., Courcelle, The monadic second order logic of graphs. I. Recognizable sets of finite graphs, Information and Computation 85 (1) (1990), 12–75.Google Scholar

[58] W. H., Cunningham, Decomposition of directed graphs, SIAM Journal on Algebraic and Discrete Methods 3 (2) (1982), 214–228.Google Scholar

[59] W. H., Cunningham and J., Edmonds, A combinatorial decomposition theory, Canadian Journal of Mathematics 32 (3) (1980), 734–765.Google Scholar

[60] E., Dahlhaus, Parallel algorithms for hierarchical clustering and applications to split decomposition and parity graph recognition, Journal of Algorithms 36 (2) (2000), 205–240.Google Scholar

[61] M. V. G., da Silva and K., Vuškovic, Triangulated neighborhoods in even-hole-free graphs, Discrete Mathematics 307 (2007), 1065–1073.

[62] M. V. G, da Silva and K., Vuškovic, Decomposition of even-holefree graphs with star cutsets and 2-joins, to appear in Journal of Combinatorial Theory Series B.

[63] C. M. H., de Figueiredo and F., Maffray, Optimizing bull-free perfect graphs, SIAM Journal on Discrete Mathematics 18 (2004), 226–240.Google Scholar

[64] C. M. H., de Figueiredo and K., Vuškovic, A class of β-perfect graphs, Discrete Mathematics 216 (2000), 169–193.Google Scholar

[65] G. A., Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71–76.Google Scholar

[66] G. A., Dirac, Minimally 2-connected graphs, Journal für die Reine und Angewandte Mathematik 228 (1967), 204–216.Google Scholar

[67] J., Edmonds, Maximum matching and a polytope with 0,1-vertices, J. Res. Nat. Bur. Standards 69B (1965), 125–130.Google Scholar

[68] J., Edmonds, Paths, trees and flowers, Canad. J. Math. 17 (1965), 449–467.Google Scholar

[69] P., Erdős and A., Hajnal, Ramsey-type theorems, Discrete Applied Mathematics 25 (1989), 37–52.Google Scholar

[70] M., Farber, On diameters and radii of bridged graphs, Discrete Mathematics 73 (1989), 249–260.Google Scholar

[71] C. P., Gabor, W. L., Hsu, and K. J., Supowit, Recognizing circle graphs in polynomial time, Journal of the ACM 36 (3) (1989), 435–473.Google Scholar

[72] T., Gallai, Graphen mit triangulierbaren ungeraden Vielecken, Magyar Tud. Akad. Mat. Kutado Int. Közl. 7 (1962), 3–36. English translation given by F. Maffray and M. Preissmann in Perfect Graphs, J. L. Ramírez-Alfonsín and B. A. Reed (editors), Willey (2001), 25–66.Google Scholar

[73] M., Garey and D. S., Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, W. H. Freeman, San Francisco (1979).

[74] J., Geelen, B., Gerards, and G., Whittle, Towards a matroid-minor structure theory, Combinatorics, Complexity and Chance, A tribute to Dominic Welsh, (eds. G., Grimmett and C., McDiarmid), Oxford University Press (2007), 72–82.

[75] R., Giles, L. E., Trotter, and A., Tucker, The strong perfect graph theorem for a class of partition able graphs, in Topics on Perfect Graphs (C., Berge and V., Chvátal, eds.), North Holland, Amsterdam (1984), 161–167.

[76] M., Grötschel, L., Lovász, and A., Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), 169–197.Google Scholar

[77] M., Grötschel, L., Lovász, and A., Schrijver, Geometric algorithms and combinatorial optimization, Springer Verlag (1988).

[78] R. B., Hayward, Weakly triangulated graphs, Journal of Combinatorial Theory Series B 39 (1985), 200–209.Google Scholar

[79] P. J., Heawood, Map colour theorem, Quart J Pure Appl Math 24 (1890), 332–338.Google Scholar

[80] A., Hertz, A fast algorithm for colouring Meyniel graphs, Journal of Combinatorial Theory Series B 50 (1990), 231–240.Google Scholar

[81] R., Hayward, C. T., Hoàng and F., Maffray, Optimizing weakly triangulated graphs, Graphs and Combinatorics 5 (1989), 339–349. See erratum in vol. 6 (1990) 33–35.Google Scholar

[82] C. T., Hoàng, Efficient algorithms for minimum weighted colouring of some classes of perfect graphs, Discrete Applied Mathematics 55 (1994), 133–143.Google Scholar

[83] W. L., Hsu, Effcient algorithms for some packing and covering problems on graphs, Ph.D. Dissertation, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, (1980).

[84] W. L., Hsu and G. L., Nemhauser, Algorithms for maximum weighted cliques, minimum weighted clique covers, and minimum colorings of claw-free perfect graphs, in Topics on Perfect Graphs (eds. C., Berge and V., Chvátal), North-Holland, Amsterdam (1984), 357–369.

[85] T., Kloks, H., Müller, and K., Vuškovic, Even-hole-free graphs that do not contain diamonds: a structure theorem and its consequences, Journal of Combinatorial Theory Series B 99 (2009), 733–800.Google Scholar

[86] B., Lévêque, D. Y., Lin, F., Maffray, and N., Trotignon, Detecting induced subgraphs, Discrete Applied Mathematics 157 (2009), 3540–3551.Google Scholar

[87] B., Lévêque, F., Maffray, and N., Trotignon, On graphs with no induced subdivision of K4, Journal of Combionatorial Theory Series B 102 (2012), 924–947.Google Scholar

[88] X., Li and W., Zang, A combinatorial algorithm for minimum weighted coloring of claw-free perfect graphs, Journal of Combinatorial Optimization 9 (4) (2005), 331–347.Google Scholar

[89] L., Lovász, Normal hypergraphs and the perfect graph conjecture, Discrete Mathematics 2 (1972), 253–267.Google Scholar

[90] L., Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25 (1979), 1–7.Google Scholar

[91] T.-H., Ma and J., Spinrad, An O(n2) algorithm for undirected split decomposition, Journal of Algorithms 16 (1) (1994), 154–160.Google Scholar

[92] F., Maffray and M., Preissmann, On the NP-completeness of the kcolorability problem for triangle-free graphs, Discrete Math. 162 (1996), 313–317.Google Scholar

[93] F., Maffray and B., Reed, A description of claw-free perfect graphs, Journal of Combinatorial Theory Series B 75 (1999), 134–156.Google Scholar

[94] F., Maffray and N., Trotignon, Algorithms for perfectly contractile graphs, SIAM Journal on Discrete Mathematics 19 (3) (2005), 553–574.Google Scholar

[95] F., Maffray, N., Trotignon, and K., Vuškovic, Algorithms for square-3PC(·, ·)-free graphs, SIAM Journal on Discrete Math 22 (1) (2008), 51–71.Google Scholar

[96] S. E., Markossian, G. S., Gasparian, and B. A., Reed, β-perfect graphs, Journal of Combinatorial Theory Series B 67 (1996), 1–11.Google Scholar

[97] T., McKee, Independent separator graphs, Utilitas Mathematica 73 (2007), 217–224.Google Scholar

[98] G. J., Minty, On maximal independent sets of vertices in claw-free graphs, Journal of Combinatorial Theory Series B 28 (1980), 284–304.Google Scholar

[99] D., Nakamura and A., Tamura, A revision of Minty's algorithm for finding a maximum weighted stable set of a claw-free graph, Journal of Operations Research Society of Japan 44 (2) (2001), 194–204.Google Scholar

[100] G., Naves, personal communication (2009).

[101] K. R., Parthasarathy and G., Ravindra, The strong perfect graph conjecture is true for K1, 3-free graphs, Journal of Combinatorial Theory Series B 21 (1976), 212–223.Google Scholar

[102] I., Penev, Forbidden substructures in graphs and trigraphs, and related coloring problems, PhD thesis, Columbia University (2012).

[103] J., Petersen, Sur le théorèm de Tait, L'Intermédiaire Math 5 (1898), 225–227.Google Scholar

[104] M. D., Plummer, On minimal blocks, Transactions of the American Mathematical Society 134 (1968), 85–94.Google Scholar

[105] S., Poljak, A note on the stable sets and coloring of graphs, Comment. Math. Univ. Carolin. 15 (1974), 307–309.Google Scholar

[106] M., Rao, Solving some NP-complete problems using split decomposition, Discrete Applied Mathematics 156 (14) (2008), 2768–2780.Google Scholar

[107] B., Reed and N., Sbihi, Recognizing bull-free perfect graphs, Graphs and Combinatorics 11 (1995), 171–178.Google Scholar

[108] N., Robertson and P. D., Seymour, Graph minors III. Planar treewidth, Journal of Combinatorial Theory Series B 36 (1984), 49–64.Google Scholar

[109] N., Robertson and P. D., Seymour, Graph minors V. Excluding a planar graph, Journal of Combinatorial Theory Series B 41 (1986), 92–114.Google Scholar

[110] N., Robertson and P. D., Seymour, Graph minors XVI. Excluding a non-planar graph, Journal of Combinatorial Theory Series B 89 (2003), 43–76.Google Scholar

[111] N., Robertson and P. D., Seymour, Graph minors XX. Wagner's Conjecture, Journal of Combinatorial Theory Series B 92 (2) (2004), 325–357.Google Scholar

[112] D. J., Rose, R. E., Tarjan and G. S., Leuker, Algorithmic aspects of vertex elimination on graphs, SIAM J. Comput. 5 (1976), 266–283.Google Scholar

[113] F., Roussel and I., Rusu, Holes and dominoes in Meyniel graphs, Internat. J. Found. Comput. Sci. 10 (1999), 127–146.Google Scholar

[114] F., Roussel and I., Rusu, An O(n2) algorithm to color Meyniel graphs, Discrete Mathematics 235 (1–3) (2001), 107–123.Google Scholar

[115] P., Seymour, Decomposition of regular matroids, Journal of Combinatorial Theory Series B 28 (1980), 305–359.Google Scholar

[116] J., Spinrad, Recognition of circle graphs, Journal of Algorithms 16 (2) (1994), 264–282.Google Scholar

[117] R. E., Tarjan, Decomposition by clique separators, Discrete Mathematics 55 (1985), 221–232.Google Scholar

[118] N., Trotignon, Decomposing Berge graphs and detecting balanced skew partitions, Journal of Combinatorial Theory Series B 98 (2008), 173–225.Google Scholar

[119] N., Trotignon and K., Vuškovic, A structure theorem for graphs with no cycle with a unique chord and its consequences, Journal of Graph Theory 63 (1) (2010), 31–67.Google Scholar

[120] N., Trotignon and K., Vuškovic, Combinatorial optimization with 2joins, Journal of Combinatorial Theory Series B 102 (2012), 153–185.Google Scholar

[121] K., Truemper, Alpha-balanced graphs and matrices and GF(3) representability of matroids, Journal of Combinatorial Theory Series B 32 (1982), 112–139.Google Scholar

[122] K., Truemper, A decomposition theory for matroids. V. Testing of matrix total unimodularity, Journal of Combinatorial Theory Series B 49 (1990), 241–281.Google Scholar

[123] K., Truemper, Matroid Decomposition, Academic Press, Boston (1992).

[124] W. T., Tutte, A homotopy theorem for matroids I, II, Transactions of the American Mathematical Society 88 (1958), 144–174.Google Scholar

[125] K., Vuškovic, Even-hole-free graphs: a survey, Applicable Analysis and Discrete Mathematics 4 (2) (2010), 219–240.Google Scholar

[126] S. H., Whitesides, An algorithm for finding clique cut-sets, Information Processing Letters 12 (1981), 31–32.Google Scholar

[127] G., Zambelli, On perfect graphs and balanced matrices, PhD thesis, Carnegie Mellon University (2004).

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