Delineating Half-Integrality of the Erdős-Pósa Property for Minors: The Case of Surfaces

Authors Christophe Paul , Evangelos Protopapas , Dimitrios M. Thilikos , Sebastian Wiederrecht



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.114.pdf
  • Filesize: 1.3 MB
  • 19 pages

Document Identifiers

Author Details

Christophe Paul
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France
Evangelos Protopapas
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France
Dimitrios M. Thilikos
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France
Sebastian Wiederrecht
  • Discrete Mathematics Group, Institute for Basic Science, Daejeon, South Korea

Cite AsGet BibTex

Christophe Paul, Evangelos Protopapas, Dimitrios M. Thilikos, and Sebastian Wiederrecht. Delineating Half-Integrality of the Erdős-Pósa Property for Minors: The Case of Surfaces. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 114:1-114:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.114

Abstract

In 1986 Robertson and Seymour proved a generalization of the seminal result of Erdős and Pósa on the duality of packing and covering cycles: A graph has the Erdős-Pósa property for minors if and only if it is planar. In particular, for every non-planar graph H they gave examples showing that the Erdős-Pósa property does not hold for H. Recently, Liu confirmed a conjecture of Thomas and showed that every graph has the half-integral Erdős-Pósa property for minors. Liu’s proof is non-constructive and to this date, with the exception of a small number of examples, no constructive proof is known. In this paper, we initiate the delineation of the half-integrality of the Erdős-Pósa property for minors. We conjecture that for every graph H, there exists a unique (up to a suitable equivalence relation on graph parameters) graph parameter EP_H such that H has the Erdős-Pósa property in a minor-closed graph class 𝒢 if and only if sup{EP_H(G) ∣ G ∈ 𝒢} is finite. We prove this conjecture for the class ℋ of Kuratowski-connected shallow-vortex minors by showing that, for every non-planar H ∈ ℋ, the parameter EP_H(G) is precisely the maximum order of a Robertson-Seymour counterexample to the Erdős-Pósa property of H which can be found as a minor in G. Our results are constructive and imply, for the first time, parameterized algorithms that find either a packing, or a cover, or one of the Robertson-Seymour counterexamples, certifying the existence of a half-integral packing for the graphs in ℋ.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Graph theory
Keywords
  • Erdős-Pósa property
  • Erdős-Pósa pair
  • Graph parameters
  • Graph minors
  • Universal obstruction
  • Surface containment

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Colin C. Adams. The knot book. American Mathematical Society, Providence, RI, 2004. An elementary introduction to the mathematical theory of knots, Revised reprint of the 1994 original. Google Scholar
  2. Isolde Adler, Stavros G. Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Irrelevant vertices for the planar disjoint paths problem. Journal of Combinatorial Theory, Series B, 122:815-843, 2017. URL: https://doi.org/10.1016/j.jctb.2016.10.001.
  3. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 951-970, 2020. URL: https://doi.org/10.1137/1.9781611975994.57.
  4. Hans L. Bodlaender. On disjoint cycles. International Journal of Foundations of Computer Science, 5(1):59-68, 1994. URL: https://doi.org/10.1142/S0129054194000049.
  5. Rutger Campbell, J. Pascal Gollin, Kevin Hendrey, and Sebastian Wiederrecht. Bipartite treewidth - The structure of non-zero cycles in group-labelled graphs, 2023, Manuscript submitted to SODA 2024, private communication. Google Scholar
  6. Radu Curticapean and Mingji Xia. Parameterizing the permanent: Hardness for fixed excluded minors. In Symposium on Simplicity in Algorithms (SOSA), pages 297-307. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977066.23.
  7. Michael J. Dinneen. Too many minor order obstructions (for parameterized lower ideals). JUCS - Journal of Universal Computer Science, 3(11):1199-1206, 1997. URL: https://doi.org/10.3217/jucs-003-11-1199.
  8. Walther Dyck. Beiträge zur Analysis situs. Mathematische Annalen, 32(4):457-512, 1888. URL: https://doi.org/10.1007/BF01443580.
  9. P. Erdős and L. Pósa. On independent circuits contained in a graph. Canadian Journal of Mathematics, 17:347-352, 1965. URL: https://doi.org/10.4153/CJM-1965-035-8.
  10. Michael R Fellows and Michael A Langston. Nonconstructive tools for proving polynomial-time decidability. Journal of the ACM, 35(3):727-739, 1988. URL: https://doi.org/doi.org/10.1145/44483.44491.
  11. Erica Flapan, Allison Henrich, Aaron Kaestner, and Sam Nelson. Knots, links, spatial graphs, and algebraic invariants, volume 689 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 2017. URL: https://doi.org/10.1090/conm/689.
  12. Fedor V Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Hitting topological minors is FPT. In ACM SIGACT Symposium on Theory of Computing (STOC), pages 1317-1326, 2020. URL: https://doi.org/10.1145/3357713.3384318.
  13. Cyril Gavoille and Claire Hilaire. Minor-universal graph for graphs on surfaces, 2023. URL: https://arxiv.org/abs/2305.06673.
  14. Jim Geelen, Bert Gerards, Bruce Reed, Paul Seymour, and Adrian Vetta. On the odd-minor variant of hadwiger’s conjecture. Journal of Combinatorial Theory, Series B, 99(1):20-29, 2009. URL: https://doi.org/10.1016/j.jctb.2008.03.006.
  15. Petr A. Golovach, Giannos Stamoulis, and Dimitrios M. Thilikos. Hitting topological minor models in planar graphs is fixed parameter tractable. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 931-950, 2020. URL: https://doi.org/10.1137/1.9781611975994.56.
  16. Petr A. Golovach, Giannos Stamoulis, and Dimitrios M. Thilikos. Combing a linkage in an annulus, 2022. URL: https://arxiv.org/abs/2207.04798.
  17. Petr A. Golovach, Giannos Stamoulis, and Dimitrios M. Thilikos. Hitting topological minor models in planar graphs is fixed parameter tractable. ACM Transactions on Algorithms, 19(3):23:1-29, 2023. URL: https://doi.org/10.1145/3583688.
  18. Tony Huynh, Felix Joos, and Paul Wollan. A unified Erdős-Pósa theorem for constrained cycles. Combinatorica, 39(1):91-133, 2019. URL: https://doi.org/10.1007/s00493-017-3683-z.
  19. Ken ichi Kawarabayashi, Robin Thomas, and Paul Wollan. Quickly excluding a non-planar graph, 2021. URL: https://arxiv.org/abs/2010.12397.
  20. Adam S. Jobson and André E. Kézdy. All minor-minimal apex obstructions with connectivity two. Electronic Journal of Combinatorics, 28(1):1.23, 58, 2021. URL: https://doi.org/10.37236/8382.
  21. Naonori Kakimura, Ken-ichi Kawarabayashi, and Dániel Marx. Packing cycles through prescribed vertices. Journal of Combinatorial Theory, Series B, 101(5):378-381, 2011. URL: https://doi.org/10.1016/j.jctb.2011.03.004.
  22. Ken-ichi Kawarabayashi. Half integral packing, Erdős-Pósa-property and graph minors. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1187-1196, 2007. URL: dl.acm.org/citation.cfm?id=1283383.1283511.
  23. Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Bruce A. Reed. The disjoint paths problem in quadratic time. Journal of Combinatorial Theory, Series B, 102(2):424-435, 2012. URL: https://doi.org/10.1016/j.jctb.2011.07.004.
  24. Ken-ichi Kawarabayashi and Stephan Kreutzer. The directed grid theorem. In ACM Symposium on Theory of Computing (STOC), pages 655-664, 2015. URL: https://doi.org/10.1145/2746539.2746586.
  25. Max Lipton, Eoin Mackall, Thomas W. Mattman, Mike Pierce, Samantha Robinson, Jeremy Thomas, and Ilan Weinschelbaum. Six variations on a theme: almost planar graphs. Involve. A Journal of Mathematics, 11(3):413-448, 2018. URL: https://doi.org/10.2140/involve.2018.11.413.
  26. Chun-Hung Liu. Packing topological minors half-integrally. Journal of the London Mathematical Society, 106(3):2193-2267, 2022. URL: https://doi.org/10.1112/jlms.12633.
  27. Dániel Marx and Ildikó Schlotter. Obtaining a planar graph by vertex deletion. Algorithmica, 62(3-4):807-822, 2012. URL: https://doi.org/10.1007/s00453-010-9484-z.
  28. Thomas W. Mattman. Forbidden minors: finding the finite few. In A primer for undergraduate research, Found. Undergrad. Res. Math., pages 85-97. Birkhäuser/Springer, Cham, 2017. Google Scholar
  29. Laure Morelle, Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. Faster Parameterized Algorithms for Modification Problems to Minor-Closed Classes. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261 of Leibniz International Proceedings in Informatics (LIPIcs), pages 93:1-93:19, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.93.
  30. Sergey Norin, Robin Thomas, and Hein van der Holst. On 2-cycles of graphs. Journal of Combinatorial Theory. Series B, 162:184-222, 2023. URL: https://doi.org/10.1016/j.jctb.2023.06.003.
  31. Christophe Paul, Evangelos Protopapas, and Dimitrios M. Thilikos. Graph parameters, universal obstructions, and wqo, 2023. URL: https://arxiv.org/abs/2304.03688.
  32. Christophe Paul, Evangelos Protopapas, and Dimitrios M. Thilikos. Universal obstructions of graph parameters, 2023. URL: https://arxiv.org/abs/2304.14121.
  33. Jean-Florent Raymond and Dimitrios M. Thilikos. Recent techniques and results on the Erdős-Pósa property. Discrete Applied Mathematics, 231:25-43, 2017. URL: https://doi.org/10.1016/j.dam.2016.12.025.
  34. Bruce Reed. Mangoes and blueberries. Combinatorica, 19(2):267-296, 1999. URL: https://doi.org/10.1007/s004930050056.
  35. Bruce Reed, Neil Robertson, Paul Seymour, and Robin Thomas. Packing directed circuits. Combinatorica, 16(4):535-554, 1996. URL: https://doi.org/10.1007/BF01271272.
  36. Neil Robertson and P. D. Seymour. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92-114, 1986. URL: https://doi.org/10.1016/0095-8956(86)90030-4.
  37. Neil Robertson and P. D. Seymour. Graph minors. XX. Wagner’s conjecture. Journal of Combinatorial Theory, Series B, 92(2):325-357, 2004. URL: https://doi.org/10.1016/j.jctb.2004.08.001.
  38. Neil Robertson, Paul Seymour, and Robin Thomas. Sach’s linkless embedding conjecture. Journal of Combinatorial Theory, Series B, 64(2):185-227, 1995. URL: https://doi.org/10.1006/jctb.1995.1032.
  39. Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. k-apices of minor-closed graph classes. II. Parameterized algorithms. ACM Transactions on Algorithms, 18(3):Art. 21, 30, 2022. URL: https://doi.org/10.1145/3519028.
  40. Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. k-apices of minor-closed graph classes. I. Bounding the obstructions. Journal of Combinatorial Theory, Series B, 161:180-227, 2023. URL: https://doi.org/10.1016/j.jctb.2023.02.012.
  41. Dimitrios M. Thilikos and Sebastian Wiederrecht. Killing a vortex. In IEEESymposium on Foundations of Computer Science (FOCS), pages 1069-1080, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00104.
  42. Dimitrios M. Thilikos and Sebastian Wiederrecht. Approximating branchwidth on parametric extensions of planarity, 2023. URL: https://arxiv.org/abs/2304.04517.
  43. Dimitrios M. Thilikos and Sebastian Wiederrecht. Excluding surfaces as minors in graphs, 2023. URL: https://arxiv.org/abs/2306.01724.
  44. Robin Thomas. Well-quasi-ordering infinite graphs with forbidden finite planar minor. Transactions of the American Mathematical Society, 312(1):279-313, 1989. URL: https://doi.org/10.2307/2001217.
  45. Wouter Cames Van Batenburg, Tony Huynh, Gwenaël Joret, and Jean-Florent Raymond. A tight Erdős-Pósa function for planar minors. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1485-1500, 2019. URL: https://doi.org/10.1137/1.9781611975482.90.
  46. Hein van der Holst. A polynomial-time algorithm to find a linkless embedding of a graph. Journal of Combinatorial Theory, Series B, 99(2):512-530, 2009. URL: https://doi.org/10.1016/j.jctb.2008.10.002.
  47. Yaming Yu. More forbidden minors for Wye-Delta-Wye reducibility. Electronic Journal of Combinatorics, 13(1):7:15, 2006. URL: https://doi.org/10.37236/1033.