Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
167 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

The grid-minor theorem revisited (2307.02816v1)

Published 6 Jul 2023 in math.CO and cs.DM

Abstract: We prove that for every planar graph $X$ of treedepth $h$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of treewidth at most $f(h)$ such that $G$ is isomorphic to a subgraph of $H\boxtimes K_c$. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the optimal parameter in such a result. As an example application, we use this result to improve the upper bound for weak coloring numbers of graphs excluding a fixed graph as a minor.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (42)
  1. Thomas Andreae. On a pursuit game played on graphs for which a minor is excluded. J. Combin. Theory, Series B, 41(1):37–47, 1986.
  2. Asymptotically optimal vertex ranking of planar graphs, 2023. arXiv:2007.06455.
  3. Product structure of graph classes with bounded treewidth, 2022. arXiv:2206.02395.
  4. Polynomial bounds for the grid-minor theorem. J. ACM, 63(5), 2016. arXiv:1305.6577.
  5. Towards tight(er) bounds for the excluded grid theorem. J. Combin. Theory, Series B, 146:219–265, 2021. arXiv:1901.07944.
  6. Some results on tree decomposition of graphs. J. Graph Theory, 20(4):481–499, 1995.
  7. Improved product structure for graphs on surfaces. Discrete Math. Theor. Comput. Sci., 24(2):#6, 2022. arXiv:2112.10025.
  8. Adjacency labelling for planar graphs (and beyond). J. ACM, 68(6):42, 2021. arXiv:2003.04280.
  9. Clustered 3-colouring graphs of bounded degree. Combin. Probab. Comput., 31(1):123–135, 2022. arXiv:2002.11721.
  10. Proof of the clustered Hadwiger conjecture, 2023. arXiv:2306.06224.
  11. Planar graphs have bounded queue-number. J. ACM, 67(4), 2020. arXiv:1904.04791.
  12. Planar graphs have bounded nonrepetitive chromatic number. Advances in Combinatorics, #5, 2020. arXiv:1904.05269.
  13. The excluded tree minor theorem revisited, 2023. arXiv:2303.14970.
  14. Zdeněk Dvořák. Constant-factor approximation of the domination number in sparse graphs. European J. Combin., 34(5):833–840, 2013. arXiv:1110.5190.
  15. Improved bounds for centered colorings. Advances in Combinatorics, #8, 2021. arXiv:1907.04586.
  16. Colouring planar graphs with three colours and no large monochromatic components. Combin., Probab. Comput., 23(4):551–570, 2014. arXiv:1303.2487.
  17. Sparse universal graphs for planarity. J. London Math. Soc., to appear. arXiv:2010.05779.
  18. Connected treewidth and connected graph searching. In José R. Correa, Alejandro Hevia, and Marcos A. Kiwi, editors, Proc. 7th Latin American Symposium on Theoretical Informatics, LATIN 2006, volume 3887 of Lecture Notes in Comput. Sci., pages 479–490. Springer, 2006.
  19. Approximating pathwidth for graphs of small treewidth. ACM Trans. Algorithms, 19(2), 2023. arXiv:2008.00779.
  20. Coloring and covering nowhere dense graphs. SIAM J. Discrete Math., 32(4):2467–2481, 2018. arXiv:1602.05926.
  21. On the generalised colouring numbers of graphs that exclude a fixed minor. European J. Combin., 66:129–144, 2017. arXiv:1602.09052.
  22. Excluding a ladder. Combinatorica, 42(3):405–432, 2022. arXiv:2002.00496.
  23. Product structure of graphs with an excluded minor, 2022. arXiv:2104.06627.
  24. Ken-ichi Kawarabayashi. Rooted minor problems in highly connected graphs. Discrete Mathematics, 287(1-3):121–123, 2004.
  25. Chun-Hung Liu. Defective coloring is perfect for minors, 2022. arXiv:2208.10729.
  26. Partitioning H𝐻Hitalic_H-minor free graphs into three subgraphs with no large components. J. Combin. Theory, Series B, 128:114–133, 2018. arXiv:1503.08371.
  27. Clustered coloring of graphs excluding a subgraph and a minor, 2019. arXiv:1905.09495.
  28. Clustered graph coloring and layered treewidth, 2019. arXiv:1905.08969.
  29. Clustered variants of Hajós’ conjecture. J. Combin. Theory, Series B, 152:27–54, 2022. arXiv:1908.05597.
  30. Graphs on surfaces. Johns Hopkins University Press, 2001.
  31. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012.
  32. Clustered colouring in minor-closed classes. Combinatorica, 39(6):1387–1412, 2019. arXiv:1708.02370.
  33. Defective colouring of graphs excluding a subgraph or minor. Combinatorica, 39(2):377–410, 2019. arXiv:1611.09060.
  34. Lecture notes for the course “Sparsity” given at Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, Winter semesters 2017/18 and 2019/20. https://www.mimuw.edu.pl/~mp248287/sparsity2.
  35. Polynomial bounds for centered colorings on proper minor-closed graph classes. J. Combin. Theory, Series B, 151:111–147, 2021. arXiv:1807.03683.
  36. On space efficiency of algorithms working on structural decompositions of graphs. ACM Trans. Computation Theory, 9(4), 2018. arXiv:1509.05896.
  37. Graph minors. I. Excluding a forest. J. Combin. Theory, Series B, 35(1):39–61, 1983.
  38. Graph minors. V. Excluding a planar graph. J. Combin. Theory, Series B, 41(1):92–114, 1986.
  39. Graph minors. XIII. The disjoint paths problem. J. Combin. Theory, Series B, 63(1):65–110, 1995.
  40. Graph minors. XVI. Excluding a non-planar graph. J. Combin. Theory, Series B, 89(1):43–76, 2003.
  41. Jan van den Heuvel and David R. Wood. Improper colourings inspired by Hadwiger’s conjecture. J. London Math. Soc., 98:129–148, 2018. arXiv:1704.06536.
  42. David R. Wood. Defective and clustered graph colouring. Electron. J. Combin., DS23, 2018. Version 1, https://www.combinatorics.org/DS23.
Citations (4)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now