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A tame vs. feral dichotomy for graph classes excluding an induced minor or induced topological minor (2405.15543v1)

Published 24 May 2024 in math.CO, cs.DM, and cs.DS

Abstract: A minimal separator in a graph is an inclusion-minimal set of vertices that separates some fixed pair of nonadjacent vertices. A graph class is said to be tame if there exists a polynomial upper bound for the number of minimal separators of every graph in the class, and feral if it contains arbitrarily large graphs with exponentially many minimal separators. Building on recent works of Gartland and Lokshtanov [SODA 2023] and Gajarsk\'y, Jaffke, Lima, Novotn\'a, Pilipczuk, Rz\k{a}.zewski, and Souza [arXiv, 2022], we show that every graph class defined by a single forbidden induced minor or induced topological minor is either tame or feral, and classify the two cases. This leads to new graph classes in which Maximum Weight Independent Set and many other problems are solvable in polynomial time. We complement the classification results with polynomial-time recognition algorithms for the maximal tame graph classes appearing in the obtained classifications.

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References (64)
  1. Graphs with polynomially many minimal separators. J. Comb. Theory, Ser. B, 152:248–280, 2022. doi:10.1016/J.JCTB.2021.10.003.
  2. Induced subgraphs of bounded treewidth and the container method. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 1948–1964. SIAM, 2021. doi:10.1137/1.9781611976465.116.
  3. Induced minor free graphs: Isomorphism and clique-width. Algorithmica, 80(1):29–47, 2018. doi:10.1007/S00453-016-0234-8.
  4. Separability generalizes Dirac’s theorem. Discrete Applied Mathematics, 84(1-3):43–53, 1998. doi:10.1016/S0166-218X(98)00005-5.
  5. Induced minors and well-quasi-ordering. J. Combin. Theory Ser. B, 134:110–142, 2019. doi:10.1016/j.jctb.2018.05.005.
  6. Maximum independent set when excluding an induced minor: K1+t⁢K2subscript𝐾1𝑡subscript𝐾2K_{1}+tK_{2}italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_t italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and t⁢C3⊎C4⊎𝑡subscript𝐶3subscript𝐶4tC_{3}\uplus C_{4}italic_t italic_C start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊎ italic_C start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 23:1–23:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. doi:10.4230/LIPICS.ESA.2023.23.
  7. V. Bouchitté and I. Todinca. Treewidth and minimum fill-in: grouping the minimal separators. SIAM Journal on Computing, 31(1):212–232, 2001. doi:10.1137/S0097539799359683.
  8. Listing all potential maximal cliques of a graph. Theoret. Comput. Sci., 276(1-2):17–32, 2002. doi:10.1016/S0304-3975(01)00007-X.
  9. Graph classes: a survey, volume 3. Philadelphia, PA: SIAM, 1999. doi:10.1137/1.9780898719796.
  10. The pigs full monty – A floor show of minimal separators. In Volker Diekert and Bruno Durand, editors, STACS 2005, 22nd Annual Symposium on Theoretical Aspects of Computer Science, Stuttgart, Germany, February 24-26, 2005, Proceedings, volume 3404 of Lecture Notes in Computer Science, pages 521–532. Springer, 2005. doi:10.1007/978-3-540-31856-9\_43.
  11. The three-in-a-tree problem. Combinatorica, 30(4):387–417, 2010. doi:10.1007/S00493-010-2334-4.
  12. Detecting an induced net subdivision. J. Comb. Theory, Ser. B, 103(5):630–641, 2013. doi:10.1016/J.JCTB.2013.07.005.
  13. Treewidth versus clique number. I. Graph classes with a forbidden structure. SIAM J. Discrete Math., 35(4):2618–2646, 2021. doi:10.1137/20M1352119.
  14. Detecting K2,3subscript𝐾23K_{2,3}italic_K start_POSTSUBSCRIPT 2 , 3 end_POSTSUBSCRIPT as an induced minor. In Adele Rescigno and Ugo Vaccaro, editors, Proceedings of the 35th International Workshop on Combinatorial Algorithms (IWOCA 2024), 2024. Lecture Notes in Computer Science, to appear.
  15. Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure. Journal of Combinatorial Theory, Series B, 167:338–391, 2024. doi:https://doi.org/10.1016/j.jctb.2024.03.005.
  16. On the vertex ranking problem for trapezoid, circular-arc and other graphs. Discrete Appl. Math., 98(1-2):39–63, 1999. doi:10.1016/S0166-218X(99)00179-1.
  17. Guoli Ding. Subgraphs and well-quasi-ordering. J. Graph Theory, 16(5):489–502, 1992. doi:10.1002/jgt.3190160509.
  18. G. A. Dirac. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25:71–76, 1961. doi:10.1007/BF02992776.
  19. Huib Donkers and Bart M. P. Jansen. A Turing kernelization dichotomy for structural parameterizations of ℱℱ{\cal F}caligraphic_F-minor-free deletion. J. Comput. System Sci., 119:164–182, 2021. doi:10.1016/j.jcss.2021.02.005.
  20. The complexity of induced minors and related problems. Algorithmica, 13(3):266–282, 1995. doi:10.1007/BF01190507.
  21. On the tractability of optimization problems on H𝐻Hitalic_H-graphs. Algorithmica, 82(9):2432–2473, 2020. doi:10.1007/s00453-020-00692-9.
  22. Algorithms parameterized by vertex cover and modular width, through potential maximal cliques. Algorithmica, 80(4):1146–1169, 2018. doi:10.1007/s00453-017-0297-1.
  23. Large induced subgraphs via triangulations and CMSO. SIAM J. Comput., 44(1):54–87, 2015. doi:10.1137/140964801.
  24. Finding induced subgraphs via minimal triangulations. In Jean-Yves Marion and Thomas Schwentick, editors, 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010, March 4-6, 2010, Nancy, France, volume 5 of LIPIcs, pages 383–394. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010. doi:10.4230/LIPICS.STACS.2010.2470.
  25. Taming graphs with no large creatures and skinny ladders. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms, ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany, volume 244 of LIPIcs, pages 58:1–58:8. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. doi:10.4230/LIPICS.ESA.2022.58.
  26. Some simplified NP-complete graph problems. Theoret. Comput. Sci., 1(3):237–267, 1976. doi:10.1016/0304-3975(76)90059-1.
  27. Peter Gartland. Quasi-Polynomial Time Techniques for Independent Set and Beyond in Hereditary Graph Classes. PhD thesis, UC Santa Barbara, 2023. https://escholarship.org/uc/item/0kk6d2jv.
  28. Independent set on Pksubscript𝑃𝑘P_{k}italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT-free graphs in quasi-polynomial time. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 613–624. IEEE, 2020. doi:10.1109/FOCS46700.2020.00063.
  29. Graph classes with few minimal separators. I. Finite forbidden induced subgraphs. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 3063–3097. SIAM, 2023. doi:10.1137/1.9781611977554.CH119.
  30. Finding large induced sparse subgraphs in C>tsubscript𝐶absent𝑡{C}_{>t}italic_C start_POSTSUBSCRIPT > italic_t end_POSTSUBSCRIPT-free graphs in quasipolynomial time. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, page 330–341, New York, NY, USA, 2021. Association for Computing Machinery. doi:10.1145/3406325.3451034.
  31. Martin Charles Golumbic. Trivially perfect graphs. Discrete Math., 24(1):105–107, 1978. doi:10.1016/0012-365X(78)90178-4.
  32. Disconnected matchings. Theoret. Comput. Sci., 956:Paper No. 113821, 17, 2023. doi:10.1016/j.tcs.2023.113821.
  33. Covering minimal separators and potential maximal cliques in Ptsubscript𝑃𝑡P_{t}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT-free graphs. Electron. J. Combin., 28(1):Paper No. 1.29, 14, 2021. doi:10.37236/9473.
  34. Polynomial-time algorithm for maximum weight independent set on P6subscript𝑃6P_{6}italic_P start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT-free graphs. ACM Trans. Algorithms, 18(1), jan 2022. doi:10.1145/3414473.
  35. On the maximum weight minimal separator. Theoret. Comput. Sci., 796:294–308, 2019. doi:10.1016/j.tcs.2019.09.025.
  36. Tatiana Romina Hartinger. New Characterizations in Structural Graph Theory: 1111-Perfectly Orientable Graphs, Graph Products, and the Price of Connectivity. PhD thesis, University of Primorska, 2017. https://www.famnit.upr.si/sl/studij/zakljucna_dela/download/532.
  37. P. Heggernes. Minimal triangulations of graphs: a survey. Discrete Mathematics, 306(3):297–317, 2006. doi:10.1016/j.disc.2005.12.003.
  38. AMP chain graphs: minimal separators and structure learning algorithms. J. Artificial Intelligence Res., 69:419–470, 2020. doi:10.1613/jair.1.12101.
  39. Ton Kloks. Treewidth of circle graphs. International Journal of Foundations of Computer Science, 7(02):111–120, 1996. doi:10.1142/S0129054196000099.
  40. Computing treewidth and minimum fill-in: All you need are the minimal separators. In Thomas Lengauer, editor, Algorithms - ESA ’93, First Annual European Symposium, Bad Honnef, Germany, September 30 - October 2, 1993, Proceedings, volume 726 of Lecture Notes in Computer Science, pages 260–271. Springer, 1993. doi:10.1007/3-540-57273-2\_61.
  41. Ekkehard Köhler. Graphs without asteroidal triples. PhD thesis, Technische Universität Berlin, 1999.
  42. Tuukka Korhonen. Finding optimal triangulations parameterized by edge clique cover. Algorithmica, 84(8):2242–2270, 2022. doi:10.1007/s00453-022-00932-0.
  43. Tuukka Korhonen. Grid induced minor theorem for graphs of small degree. J. Combin. Theory Ser. B, 160:206–214, 2023. doi:10.1016/j.jctb.2023.01.002.
  44. Induced-minor-free graphs: Separator theorem, subexponential algorithms, and improved hardness of recognition. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 5249–5275. SIAM, 2024. doi:10.1137/1.9781611977912.188.
  45. Three-in-a-tree in near linear time. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 1279–1292. ACM, 2020. doi:10.1145/3357713.3384235.
  46. Ngoc Khang Le. Detecting an induced subdivision of K4subscript𝐾4K_{4}italic_K start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT. J. Graph Theory, 90(2):160–171, 2019. doi:10.1002/jgt.22374.
  47. Detecting induced subgraphs. Discret. Appl. Math., 157(17):3540–3551, 2009. doi:10.1016/J.DAM.2009.02.015.
  48. Beyond classes of graphs with “few” minimal separators: FPT results through potential maximal cliques. Algorithmica, 81(3):986–1005, 2019. doi:10.1007/s00453-018-0453-2.
  49. Independence and efficient domination on P6subscript𝑃6P_{6}italic_P start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT-free graphs. ACM Trans. Algorithms, 14(1):Art. 3, 30, 2018. doi:10.1145/3147214.
  50. Algorithms for square-3pc(⋅,⋅⋅⋅\cdot,\cdot⋅ , ⋅)-free Berge graphs. SIAM J. Discrete Math., 22(1):51–71, 2008. doi:10.1137/050628520.
  51. Terry A. McKee. Graphs that have separator tree representations. Australas. J. Combin., 80:89–98, 2021.
  52. Polynomially bounding the number of minimal separators in graphs: Reductions, sufficient conditions, and a dichotomy theorem. Electron. J. Comb., 28(1), 2021. doi:10.37236/9428.
  53. Bojan Mohar. Face covers and the genus problem for apex graphs. J. Comb. Theory, Ser. B, 82(1):102–117, 2001. doi:10.1006/jctb.2000.2026.
  54. Minimal separators in P4subscript𝑃4P_{4}italic_P start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT-sparse graphs. Discrete Math., 306(3):381–392, 2006. doi:10.1016/j.disc.2005.12.008.
  55. Isomorphism on subgraph-closed graph classes: A complexity dichotomy and intermediate graph classes. In Leizhen Cai, Siu-Wing Cheng, and Tak Wah Lam, editors, Algorithms and Computation - 24th International Symposium, ISAAC 2013, Hong Kong, China, December 16-18, 2013, Proceedings, volume 8283 of Lecture Notes in Computer Science, pages 111–118. Springer, 2013. doi:10.1007/978-3-642-45030-3\_11.
  56. Minimal separators in extended P4subscript𝑃4P_{4}italic_P start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT-laden graphs. Discrete Appl. Math., 160(18):2769–2777, 2012. doi:10.1016/j.dam.2012.01.025.
  57. Quasi-polynomial-time algorithm for independent set in Ptsubscript𝑃𝑡P_{t}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT-free graphs via shrinking the space of induced paths. In Hung Viet Le and Valerie King, editors, 4th Symposium on Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12, 2021, pages 204–209. [Society for Industrial and Applied Mathematics (SIAM)], Philadelphia, PA, 2021. doi:10.1137/1.9781611976496.23.
  58. Robust algorithms for restricted domains. J. Algorithms, 48(1):160–172, 2003. doi:10.1016/S0196-6774(03)00048-8.
  59. Konstantin Skodinis. Efficient analysis of graphs with small minimal separators. In Peter Widmayer, Gabriele Neyer, and Stephan J. Eidenbenz, editors, Graph-Theoretic Concepts in Computer Science, 25th International Workshop, WG ’99, Ascona, Switzerland, June 17-19, 1999, Proceedings, volume 1665 of Lecture Notes in Computer Science, pages 155–166. Springer, 1999. doi:10.1007/3-540-46784-X\_16.
  60. Jeremy Spinrad. On comparability and permutation graphs. SIAM Journal on Computing, 14(3):658–670, 1985. doi:10.1137/0214048.
  61. Karol Suchan. Minimal Separators in Intersection Graphs. Master’s thesis, Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie, 2003.
  62. Hisao Tamaki. Computing treewidth via exact and heuristic lists of minimal separators. In Ilias S. Kotsireas, Panos M. Pardalos, Konstantinos E. Parsopoulos, Dimitris Souravlias, and Arsenis Tsokas, editors, Analysis of Experimental Algorithms - Special Event, SEA2 2019, Kalamata, Greece, June 24-29, 2019, Revised Selected Papers, volume 11544 of Lecture Notes in Computer Science, pages 219–236. Springer, 2019. doi:10.1007/978-3-030-34029-2\_15.
  63. Hisao Tamaki. Positive-instance driven dynamic programming for treewidth. J. Comb. Optim., 37(4):1283–1311, 2019. doi:10.1007/s10878-018-0353-z.
  64. χ𝜒\chiitalic_χ-bounds, operations, and chords. Journal of Graph Theory, 88(2):312–336, 2018. doi:10.1002/jgt.22214.
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Authors (2)
  1. Martin Milanič (85 papers)
  2. Nevena Pivač (9 papers)

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