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Graph Minors and Parameterized Algorithm Design

  • Chapter
The Multivariate Algorithmic Revolution and Beyond

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7370))

Abstract

The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic applications of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique.

This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: “Thalis. Investing in knowledge society through the European Social Fund”.

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References

  1. Adler, I., Dorn, F., Fomin, F.V., Sau, I., Thilikos, D.M.: Fast Minor Testing in Planar Graphs. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6346, pp. 97–109. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  2. Adler, I., Grohe, M., Kreutzer, S.: Computing excluded minors. In: Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, pp. 641–650. Society for Industrial and Applied Mathematics, Philadelphia (2008)

    Google Scholar 

  3. Adler, I., Kolliopoulos, S.G., Krause, P.K., Lokshtanov, D., Saurabh, S., Thilikos, D.: Tight Bounds for Linkages in Planar Graphs. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 110–121. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  4. Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alber, J., Dorn, F., Niedermeier, R.: Experimental evaluation of a tree decomposition-based algorithm for vertex cover on planar graphs. Discrete Applied Mathematics 145(2), 219–231 (2005); Structural Decompositions, Width Parameters, and Graph Labelings

    Article  MathSciNet  MATH  Google Scholar 

  6. Alber, J., Niedermeier, R.: Improved Tree Decomposition Based Algorithms for Domination-like Problems. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 613–627. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  7. Amir, E.: Approximation algorithms for treewidth. Algorithmica 56, 448–479 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. Journal of Algorithms 12, 308–340 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Betzler, N., Niedermeier, R., Uhlmann, J.: Tree decompositions of graphs: Saving memory in dynamic programming. Discrete Optimization 3(3), 220–229 (2006); Graphs and Combinatorial Optimization

    Article  MathSciNet  MATH  Google Scholar 

  10. Bienstock, D., Dean, N.: On obstructions to small face covers in planar graphs. J. Combin. Theory Ser. B 55(2), 163–189 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbius: fast subset convolution. In: STOC, pp. 67–74. ACM (2007)

    Google Scholar 

  12. Bodlaender, H., Fomin, F., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.: (Meta) kernelization. In: 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009). ACM (2009)

    Google Scholar 

  13. Bodlaender, H.L.: Dynamic Programming on Graphs with Bounded Treewidth. In: Lepistö, T., Salomaa, A. (eds.) ICALP 1988. LNCS, vol. 317, pp. 105–118. Springer, Heidelberg (1988)

    Chapter  Google Scholar 

  14. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bodlaender, H.L., Fellows, M.R., Hallett, M.T.: Beyond NP-completeness for problems of bounded width: hardness for the W hierarchy. In: Twenty-sixth Annual ACM Symposium on Theory of Computing (STOC 1994), pp. 449–458. ACM, New York (1994)

    Chapter  Google Scholar 

  16. Bodlaender, H.L., Telle, J.A.: Space-efficient construction variants of dynamic programming. Nordic J. Comput. 11(4), 374–385 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Borie, R.B., Parker, R.G., Tovey, C.A.: Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica 7, 555–581 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. J. Comput. System Sci. 67(4), 789–807 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cattell, K., Dinneen, M.J., Downey, R.G., Fellows, M.R., Langston, M.A.: On computing graph minor obstruction sets. Theor. Comput. Sci. 233, 107–127 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chlebíková, J.: The structure of obstructions to treewidth and pathwidth. Discrete Applied Mathematics 120(1-3), 61–71 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Courcelle, B., Downey, R.G., Fellows, M.R.: A note on the computability of graph minor obstruction sets for monadic second order ideals. In: First Japan-New Zealand Workshop on Logic in Computer Science, Auckland, vol. 3, pp. 1194–1198 (1997) (electronic)

    Google Scholar 

  22. Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS 2011), pp. 150–159. IEEE Computer Society (2011)

    Google Scholar 

  23. Dahan, X., Tillich, J.-P.: Ramanujan graphs of very large girth based on octonions. CoRR, arXiv:1011.2642 (November 2010-2011)

    Google Scholar 

  24. Dawar, A., Grohe, M., Kreutzer, S.: Locally excluding a minor. In: 21st IEEE Symposium on Logic in Computer Science (LICS 2007), pp. 270–279. IEEE, New York (2007)

    Chapter  Google Scholar 

  25. Dawar, A., Kreutzer, S.: Domination problems in nowhere-dense classes. In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FST-TCS 2009), pp. 157–168 (2009)

    Google Scholar 

  26. Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Bidimensional parameters and local treewidth. SIAM J. Discrete Math. 18(3), 501–511 (2004) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  27. Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs. Journal of the ACM 52(6), 866–893 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Demaine, E.D., Hajiaghayi, M.: Bidimensionality: new connections between FPT algorithms and PTASs. In: Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 590–601. ACM, New York (2005) (electronic)

    Google Scholar 

  29. Demaine, E.D., Hajiaghayi, M.: Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica 28(1), 19–36 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Demaine, E.D., Hajiaghayi, M., Kawarabayashi, K.: Algorithmic graph minor theory: Improved grid minor bounds and Wagner’s contraction. Algorithmica 54(2), 142–180 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Demaine, E.D., Hajiaghayi, M., Thilikos, D.M.: The bidimensional theory of bounded-genus graphs. SIAM J. Discrete Math. 20(2), 357–371 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Diestel, R.: Graph Theory, 3rd edn. Graduate Texts in Mathematics, vol. 173. Springer (2005)

    Google Scholar 

  33. Diestel, R., Jensen, T.R., Gorbunov, K.Y., Thomassen, C.: Highly connected sets and the excluded grid theorem. J. Combin. Theory Ser. B 75(1), 61–73 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dinneen, M.J.: Too many minor order obstructions (for parameterized lower ideals). In: First Japan-New Zealand Workshop on Logic in Computer Science, Auckland, vol. 3(11), pp. 1199–1206 (1997) (electronic)

    Google Scholar 

  35. Dorn, F.: Dynamic Programming and Fast Matrix Multiplication. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 280–291. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  36. Dorn, F., Fomin, F.V., Thilikos, D.M.: Fast Subexponential Algorithm for Non-local Problems on Graphs of Bounded Genus. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 172–183. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  37. Dorn, F., Fomin, F.V., Thilikos, D.M.: Catalan structures and dynamic programming in H-minor-free graphs. In: ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 631–640. SIAM (2008)

    Google Scholar 

  38. Dorn, F., Penninkx, E., Bodlaender, H.L., Fomin, F.V.: Efficient exact algorithms on planar graphs: Exploiting sphere cut decompositions. Algorithmica 58(3), 790–810 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Downey, R.G., Fellows, M.R.: Parameterized complexity. Monographs in Computer Science. Springer, New York (1999)

    Book  MATH  Google Scholar 

  40. Ellis, J.A., Sudborough, I.H., Turner, J.S.: The vertex separation and search number of a graph. Information and Computation 113(1), 50–79 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  41. Erdős, P., Sachs, H.: Reguläre graphen gegebener tailenweite mit minimaler knollenzahh. Wiss. Z. Univ. Halle-Willenberg Math. Nat. 12, 251–258 (1063)

    Google Scholar 

  42. Fellows, M.: Towards Fully Multivariate Algorithmics: Some New Results and Directions in Parameter Ecology. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 2–10. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  43. Fellows, M.R., Langston, M.A.: On search, decision, and the efficiency of polynomial-time algorithms. J. Comput. System Sci. 49(3), 769–779 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  44. Flum, J., Grohe, M.: Parameterized Complexity theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)

    MATH  Google Scholar 

  45. Fomin, F.V., Daniel Lokshtanov, S.S., Thilikos, D.M.: Linear kernels for (connected) dominating set on h-minor-free graphs. In: 23st ACM–SIAM Symposium on Discrete Algorithms (SODA 2012). ACM-SIAM, San Francisco (2012)

    Google Scholar 

  46. Fomin, F.V., Golovach, P.A., Thilikos, D.M.: Contraction obstructions for treewidth. J. Comb. Theory, Ser. B 101(5), 302–314 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  47. Fomin, F.V., Lokshtanov, D., Misra, N., Philip, G., Saurabh, S.: Hitting forbidden minors: Approximation and kernelization. In: STACS, pp. 189–200 (2011)

    Google Scholar 

  48. Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S.: Bidimensionality and EPTAS. In: 22st ACM–SIAM Symposium on Discrete Algorithms (SODA 2011), pp. 748–759. ACM-SIAM, San Francisco (2011)

    Google Scholar 

  49. Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S.: Subexponential algorithms for partial cover problems. Inf. Process. Lett. 111(16), 814–818 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Fomin, F.V., Lokshtanov, D., Saurabh, S.: Bidimensionality and geometric graphs. In: 23st ACM–SIAM Symposium on Discrete Algorithms (SODA 2012). ACM-SIAM, San Francisco (2012)

    Google Scholar 

  51. Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), Austin, Texas, pp. 503–510. ACM-SIAM (2010)

    Google Scholar 

  52. Fomin, F.V., Thilikos, D.M.: Dominating sets in planar graphs: branch-width and exponential speed-up. SIAM J. Comput. 36(2), 281–309 (2006) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  53. Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic 130(1-3), 3–31 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  54. Friedman, H., Robertson, N., Seymour, P.D.: The metamathematics of the graph minor theorem. In: Logic and Combinatorics (Arcata, Calif., 1985). Contemp. Math., vol. 65, pp. 229–261. Amer. Math. Soc., Providence (1987)

    Chapter  Google Scholar 

  55. Friedman, H.M.: Internal finite tree embeddings. In: Reflections on the foundations of mathematics (Stanford, CA, 1998). Lect. Notes Log., vol. 15, pp. 60–91. Assoc. Symbol. Logic, Urbana (2002)

    Google Scholar 

  56. Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Combin. Theory, Ser. B 16(1), 47–56 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  57. Geelen, J.F., Gerards, A.M.H., Robertson, N., Whittle, G.P.: On the excluded minors for the matroids of branch-width k. J. Combin. Theory Ser. B 88(2), 261–265 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  58. Giannopoulou, A.C., Thilikos, D.M.: Obstructions for tree-depth. Electronic Notes in Discrete Mathematics 34, 249–253 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  59. Giannopoulou, A.C., Thilikos, D.M.: Optimizing the graph minors weak structure theorem. CoRR, arXiv:1102.5762 (February 2011)

    Google Scholar 

  60. Golovach, P.A., Kamiński, M., Paulusma, D., Thilikos, D.M.: Induced Packing of Odd Cycles in a Planar Graph. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 514–523. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  61. Grohe, M., Kawarabayashi, K., Marx, D., Wollan, P.: Finding topological subgraphs is fixed-parameter tractable. In: Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC 2011), pp. 479–488 (2011)

    Google Scholar 

  62. Gu, Q.-P., Tamaki, H.: Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part II. LNCS, vol. 6507, pp. 85–96. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  63. Heggernes, P., van’t Hof, P., Jansen, B.M.P., Kratsch, S., Villanger, Y.: Parameterized Complexity of Vertex Deletion into Perfect Graph Classes. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 240–251. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  64. Heggernes, P., van’t Hof, P., Lokshtanov, D., Paul, C.: Obtaining a bipartite graph by contracting few edges. In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011), pp. 217–228 (2011)

    Google Scholar 

  65. Higman, G.: Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2(3), 326–336 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  66. Ito, T., Kamiński, M., Paulusma, D., Thilikos, D.M.: Parameterizing cut sets in a graph by the number of their components. Theor. Comput. Sci. 412(45), 6340–6350 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  67. Johnson, D.S.: The NP-completeness column: An ongoing guide. Journal of Algorithms 8(2), 285–303 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  68. Kamiński, M., Nishimura, N.: Finding an induced path of given parity in planar graphs in polynomial time. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2011), pp. 656–670. ACM (2012)

    Google Scholar 

  69. Kamiński, M., Thilikos, D.M.: Contraction checking in graphs on surfaces. In: 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012), pp. 182–193 (2012)

    Google Scholar 

  70. Kawarabayashi, K.: Half integral packing, Erdős-Pósa-property and graph minors. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pp. 1187–1196. Society for Industrial and Applied Mathematics, Philadelphia (2007)

    Google Scholar 

  71. Kawarabayashi, K.: Planarity allowing few error vertices in linear time. In: 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, pp. 639–648 (2009)

    Google Scholar 

  72. Kawarabayashi, K., Kobayashi, Y.: The edge disjoint paths problem in eulerian graphs and 4-edge-connected graphs. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 345–353. Society for Industrial and Applied Mathematics, Philadelphia (2010)

    Chapter  Google Scholar 

  73. Kawarabayashi, K., Kobayashi, Y.: An improved algorithm for the half-disjoint paths problem. SIAM J. Discrete Math. 25(3), 1322–1330 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  74. Kawarabayashi, K., Kobayashi, Y., Reed, B.: The disjoint paths problem in quadratic time. Journal of Combinatorial Theory, Series B (2011)

    Google Scholar 

  75. Kawarabayashi, K., Kreutzer, S., Mohar, B.: Linkless and flat embeddings in 3-space and the unknot problem. In: Proceedings of the 2010 Annual Symposium on Computational Geometry, SoCG 2010, pp. 97–106. ACM, New York (2010)

    Chapter  Google Scholar 

  76. Kawarabayashi, K., Li, Z., Reed, B.A.: Recognizing a totally odd K4-subdivision, parity 2-disjoint rooted paths and a parity cycle through specified elements. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 318–328 (2010)

    Google Scholar 

  77. Kawarabayashi, K., Mohar, B., Reed, B.A.: A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded tree-width. In: 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, pp. 771–780 (2008)

    Google Scholar 

  78. Kawarabayashi, K., Reed, B.A.: Hadwiger’s conjecture is decidable. In: 41st Annual ACM Symposium on Theory of Computing (STOC 2009), pp. 445–454 (2009)

    Google Scholar 

  79. Kawarabayashi, K., Reed, B.A.: Odd cycle packing. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, pp. 695–704 (2010)

    Google Scholar 

  80. Kawarabayashi, K., Wollan, P.: A shorter proof of the graph minor algorithm: the unique linkage theorem. In: 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, pp. 771–780 (2008)

    Google Scholar 

  81. Kneis, J., Langer, A.: A practical approach to courcelle’s theorem. Electron. Notes Theor. Comput. Sci. 251, 65–81 (2009)

    Article  MATH  Google Scholar 

  82. Kneis, J., Langer, A., Rossmanith, P.: Courcelle’s theorem - a game-theoretic approach. CoRR, arXiv:1104.3905 (April 2011)

    Google Scholar 

  83. Kobayashi, Y., Kawarabayashi, K.: Algorithms for finding an induced cycle in planar graphs and bounded genus graphs. In: 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pp. 1146–1155. ACM-SIAM (2009)

    Google Scholar 

  84. Koutsonas, A., Thilikos, D.M., Yamazaki, K.: Outerplanar obstructions for matroid pathwidth. In: EuroComb 2011: European Conference on Combinatorics, Graph Theory and Applications (2011)

    Google Scholar 

  85. Kreutzer, S., Tazari, S.: On brambles, grid-like minors, and parameterized intractability of monadic second-order logic. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 354–364. Society for Industrial and Applied Mathematics, Philadelphia (2010)

    Chapter  Google Scholar 

  86. Kruskal, J.B.: Well-quasi-ordering, the tree theorem, and Vazsonyi’s conjecture. Trans. Amer. Math. Soc. 95, 210–225 (1960)

    MathSciNet  MATH  Google Scholar 

  87. Kuratowski, K.: Sur le problème des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)

    MATH  Google Scholar 

  88. Lagergren, J.: Efficient parallel algorithms for graphs of bounded tree-width. Journal of Algorithms 20(1), 20–44 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  89. Lagergren, J.: Upper bounds on the size of obstructions and intertwines. J. Combin. Theory, Ser. B 73, 7–40 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  90. Lokshtanov, D., Marx, D., Saurabh, S.: Slightly superexponential parameterized problems. In: 22st ACM–SIAM Symposium on Discrete Algorithms (SODA 2011), pp. 760–776 (2011)

    Google Scholar 

  91. Lovász, L.: Graph minor theory. Bull. Amer. Math. Soc. (N.S.) 43(1), 75–86 (2006) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  92. Marx, D.: Chordal deletion is fixed-parameter tractable. Algorithmica 57(4), 747–768 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  93. Marx, D., Schlotter, I.: Obtaining a planar graph by vertex deletion. Algorithmica 62(3-4), 807–822 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  94. Mohar, B.: A linear time algorithm for embedding graphs in an arbitrary surface. SIAM J. Discrete Math. 12(1), 6–26 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  95. Nash-Williams, C.S.J.A.: On well-quasi-ordering finite trees. Proc. Cambridge Philos. Soc. 59, 833–835 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  96. Neumann, B.H.: On ordered division rings. Trans. Amer. Math. Soc. 66, 202–252 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  97. Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  98. Parsons, T.D.: Pursuit-evasion in a graph. In: Proceedings Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976, Theory and Applications of Graphs. Lecture Notes in Math., vol. 642, pp. 426–441. Springer, Berlin (1978)

    Google Scholar 

  99. Ramachandramurthi, S.: The structure and number of obstructions to treewidth. SIAM J. Discrete Math. 10(1), 146–157 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  100. Reed, B.A.: Finding approximate separators and computing tree width quickly. In: Twenty-Fourth Annual ACM Symposium on Theory of Computing (STOC 1992), pp. 221–228. ACM Press (1992)

    Google Scholar 

  101. Reed, B.A., Wood, D.R.: Polynomial treewidth forces a large grid-like-minor. Eur. J. Comb. 33(3), 374–379 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  102. Robertson, N., Seymour, P.D.: Graph minors. I. excluding a forest. J. Combin. Theory, Ser. B 35, 39–61 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  103. Robertson, N., Seymour, P.D.: Graph minors. III. Planar tree-width. J. Combin. Theory, Ser. B 36(1), 49–64 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  104. Robertson, N., Seymour, P.D.: Graph minors. II. algorithmic aspects of tree-width. Journal of Algorithms 7, 309–322 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  105. Robertson, N., Seymour, P.D.: Graph minors. V. Excluding a planar graph. J. Combin. Theory Ser. B 41(1), 92–114 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  106. Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Combin. Theory Ser. B 52(2), 153–190 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  107. Robertson, N., Seymour, P.D.: Graph minors. XXII. Irrelevant vertices in linkage problems (1992) (preprint)

    Google Scholar 

  108. Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Combin. Theory, Ser. B 63(1), 65–110 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  109. Robertson, N., Seymour, P.D.: Graph minors. XVI. Excluding a non-planar graph. J. Combin. Theory Series B 77, 1–27 (1999)

    Article  MATH  Google Scholar 

  110. Robertson, N., Seymour, P.D.: Graph minors. XX. Wagner’s conjecture. J. Combin. Theory Ser. B 92(2), 325–357 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  111. Robertson, N., Seymour, P.D.: Graph minors. XXI. Graphs with unique linkages. J. Combin. Theory Ser. B 99(3), 583–616 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  112. Robertson, N., Seymour, P.D.: Graph minors XXIII. Nash-Williams’ immersion conjecture. J. Combin. Theory Ser. B 100(2), 181–205 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  113. Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Combin. Theory Ser. B 62(2), 323–348 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  114. Rué, J., Sau, I., Thilikos, D.M.: Dynamic Programming for Graphs on Surfaces. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 372–383. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  115. Rué, J., Stavropoulos, K.S., Thilikos, D.M.: Outerplanar obstructions for the feedback vertex set. Electronic Notes in Discrete Mathematics 34, 167–171 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  116. Scheffler, P.: A practical linear time algorithm for disjoint paths in graphs with bounded tree-width. Technical Report 396/1994, FU Berlin, Fachbereich 3 Mathematik (1994)

    Google Scholar 

  117. Takahashi, A., Ueno, S., Kajitani, Y.: Minimal acyclic forbidden minors for the family of graphs with bounded path-width. Disc. Math. 127(1-3), 293–304 (1994); Graph theory and applications, Hakone (1990)

    Article  MathSciNet  MATH  Google Scholar 

  118. Thilikos, D.M.: Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics 105, 239–271 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  119. van Leeuwen, J.: Graph algorithms. In: Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (A), pp. 525–631. Elsevier Science (1990)

    Google Scholar 

  120. van Rooij, J.M.M., Bodlaender, H.L., Rossmanith, P.: Dynamic Programming on Tree Decompositions Using Generalised Fast Subset Convolution. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 566–577. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  121. Wagner, K.: Über eine eigenschaft der ebenen komplexe. Mathematische Annalen 114, 570–590 (1937), 10.1007/BF01594196

    Article  MathSciNet  MATH  Google Scholar 

  122. Weiss, A.: Girths of bipartite sextet graphs. Combinatorica 4(2-3), 241–245 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, GR-15784, Athens, Greece

    Dimitrios M. Thilikos

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  1. Dimitrios M. Thilikos
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  1. Department of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. School of Mathematics, Statistics and Operations Research, Victoria University, P.O. Box 600, Wellington, New Zealand

    Rod Downey

  3. Institute of Informatics, University of Bergen, Postboks 7800, 5020, Bergen, Norway

    Fedor V. Fomin

  4. Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI), Pf. 63, 1518, Budapest, Hungary

    Dániel Marx

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Thilikos, D.M. (2012). Graph Minors and Parameterized Algorithm Design. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds) The Multivariate Algorithmic Revolution and Beyond. Lecture Notes in Computer Science, vol 7370. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30891-8_13

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