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Excluding a Ladder


A ladder is a 2 × k grid graph. When does a graph class \({\cal C}\) exclude some ladder as a minor? We show that this is the case if and only if all graphs G in \({\cal C}\) admit a proper vertex coloring with a bounded number of colors such that for every 2-connected subgraph H of G, there is a color that appears exactly once in H. This type of vertex coloring is a relaxation of the notion of centered coloring, where for every connected subgraph H of G, there must be a color that appears exactly once in H. The minimum number of colors in a centered coloring of G is the treedepth of G, and it is known that classes of graphs with bounded treedepth are exactly those that exclude a fixed path as a subgraph, or equivalently, as a minor. In this sense, the structure of graphs excluding a fixed ladder as a minor resembles the structure of graphs without long paths. Another similarity is as follows: It is an easy observation that every connected graph with two vertex-disjoint paths of length k has a path of length k+1. We show that every 3-connected graph which contains as a minor a union of sufficiently many vertex-disjoint copies of a 2×k grid has a 2×(k+1) grid minor.

Our structural results have applications to poset dimension. We show that posets whose cover graphs exclude a fixed ladder as a minor have bounded dimension. This is a new step towards the goal of understanding which graphs are unavoidable as minors in cover graphs of posets with large dimension.

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  1. K. Ando, H. Enomoto and A. Saito: Contractible edges in 3-connected graphs, Journal of Combinatorial Theory, Series B 42 (1987), 87–93.

    MathSciNet  Article  Google Scholar 

  2. R. Diestel: Graph Theory, Springer Publishing Company, Incorporated, 5th edition, 2017.

  3. P. Erdős and G. Szekeres: A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470.

    MathSciNet  MATH  Google Scholar 

  4. M. Gorsky and M. T. Seweryn: Posets with k-outerplanar cover graphs have bounded dimension, arXiv:2103.15920.

  5. R. Halin: Untersuchengen über minimale n-fach zusammenhängende graphen, Mathematische Annalen 182 (1969), 175–188.

    MathSciNet  Article  Google Scholar 

  6. R. Halin: Zur theorie der n-fach zusammenhägenden graphen, Abh. Math. Sem. Univ. Hamburg 33 (1969), 133–164.

    MathSciNet  Article  Google Scholar 

  7. D. M. Howard, N. Streib, W. T. Trotter, B. Walczak and R. Wang: Dimension of posets with planar cover graphs excluding two long incomparable chains, Journal of Combinatorial Theory, Series A 164 (2019), 1–23.

    MathSciNet  Article  Google Scholar 

  8. G. Joret, P. Micek, P. O. de Mendez and V. Wiechert: Nowhere dense graph classes and dimension, Combinatorica 39 (2019), 1055–1079.

    MathSciNet  Article  Google Scholar 

  9. G. Joret, P. Micek, K. G. Milans, W. T. Trotter, B. Walczak and R. Wang: Tree-width and dimension, Combinatorica 36 (2016), 431–450.

    MathSciNet  Article  Google Scholar 

  10. G. Joret, P. Micek, W. T. Trotter, R. Wang and V. Wiechert: On the dimension of posets with cover graphs of treewidth 2, Order 34 (2017), 185–234.

    MathSciNet  Article  Google Scholar 

  11. G. Joret, P. Micek and V. Wiechert: Planar posets have dimension at most linear in their height, SIAM J. Discrete Math. 31 (2018), 2754–2790.

    MathSciNet  Article  Google Scholar 

  12. G. Joret, P. Micek and V. Wiechert: Sparsity and dimension, Combinatorica 38 (2018), 1129–1148.

    MathSciNet  Article  Google Scholar 

  13. D. Kelly: On the dimension of partially ordered sets, Discrete Mathematics 35 (1981), 135–156.

    MathSciNet  Article  Google Scholar 

  14. J. Kozik, P. Micek and W. T. Trotter: Dimension is polynomial in height for posets with planar cover graphs, arXiv:1907.00380.

  15. P. Micek and V. Wiechert: Topological minors of cover graphs and dimension, Journal of Graph Theory 86 (2017), 295–314.

    MathSciNet  Article  Google Scholar 

  16. J. Nešetřil P. O. de Mendez: Sparsity, volume 28 of Algorithms and Combinatorics, Graphs, structures, and algorithms, Springer, Heidelberg, 2012.

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  18. M. T. Seweryn: Improved bound for the dimension of posets of treewidth two, Discrete Mathematics, 343(1):111605, 2020.

    MathSciNet  Article  Google Scholar 

  19. N. Streib and W. T. Trotter: Dimension and height for posets with planar cover graphs, European J. Combin. 35 (2014), 474–489.

    MathSciNet  Article  Google Scholar 

  20. W. T. Trotter, B. Walczak and R. Wang: Dimension and cut vertices: an application of Ramsey theory, in: Connections in Discrete Mathematics, 187–199, Cambridge Univ. Press, Cambridge, 2018.

    Chapter  Google Scholar 

  21. B. Walczak: Minors and dimension, J. Combin. Theory Ser. B 122 (2017), 668–689.

    MathSciNet  Article  Google Scholar 

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We are much grateful to the three anonymous referees for their careful reading and very helpful comments.

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Correspondence to Michał T. Seweryn.

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T. Huynh is supported by the Australian Research Council. G. Joret is supported by an ARC grant from the Wallonia-Brussels Federation of Belgium. P. Micek was partially supported by the National Science Center of Poland under grant no. 2015/18/E/ST6/00299. M. T. Seweryn was partially supported by Kartezjusz program WND-POWR.03.02.00-00-I001/16-01 funded by The National Center for Research and Development of Poland.

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Huynh, T., Joret, G., Micek, P. et al. Excluding a Ladder. Combinatorica 42, 405–432 (2022).

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Mathematics Subject Classification (2010)

  • 05C83
  • 06A07