Packing and Covering Balls in Graphs Excluding a Minor

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We prove that for every integer t ⩾ 1 there exists a constant ct such that for every Kt-minor-free graph G, and every set S of balls in G, the minimum size of a set of vertices of G intersecting all the balls of S is at most ct times the maximum number of vertex-disjoint balls in S. This was conjectured by Chepoi, Estellon, and Vaxès in 2007 in the special case of planar graphs and of balls having the same radius.

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  • 06 April 2021

    The representation of an author’s name has been corrected to read “Cames van Batenburg, W.” instead of “Van Batenburg, W.C.”


  1. [1]

    G. Borradaile and E. W. Chambers: Covering Nearly Surface-Embedded Graphs with a Fixed Number of Balls, Discrete Comput. Geom. 51 (2014), 979–996.

    MathSciNet  Article  Google Scholar 

  2. [2]

    N. Bousquet and S. Thomassé: VC-dimension and Erdős-Pósa property of graphs, Discrete Math. 338 (2015), 2302–2317.

    MathSciNet  Article  Google Scholar 

  3. [3]

    P. J. Cameron: Problems from CGCS Luminy, May 2007, European J. Combin. 31 (2010), 644–648.

    MathSciNet  Article  Google Scholar 

  4. [4]

    W. Cames van Batenburg, L. Esperet and T. Müller: Coloring Jordan regions and curves, SIAM J. Discrete Math. 31 (2017), 1670–1684.

    MathSciNet  Article  Google Scholar 

  5. [5]

    Y. Caro: New results on the independence number, Technical report, Tel Aviv University, 1979.

  6. [6]

    V. Chepoi, B. Estellon and G. Naves: Packing and covering with balls on Busemann surfaces, Discrete Comput. Geom. 57 (2017), 985–1011.

    MathSciNet  Article  Google Scholar 

  7. [7]

    V. Chepoi, B. Estellon and Y. Vaxès: Covering planar graphs with a fixed number of balls, Discrete Comput. Geom. 37 (2007), 237–244.

    MathSciNet  Article  Google Scholar 

  8. [8]

    G. Ding, P. D. Seymour and P. Winkler: Bounding the vertex cover number of a hypergraph, Combinatorica 14 (1994), 23–34.

    MathSciNet  Article  Google Scholar 

  9. [9]

    G. Ducoffe, M. Habib and L. Viennot: Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension, in: Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA), 2020.

  10. [10]

    Z. Dvořák: Constant-factor approximation of the domination number in sparse graphs, European J. Combin. 34(5) (2013), 833–840.

    MathSciNet  Article  Google Scholar 

  11. [11]

    Z. Dvořák and S. Norin: Strongly sublinear separators and polynomial expansion, SIAM J. Discrete Math. 30(2) (2016), 1095–1101.

    MathSciNet  Article  Google Scholar 

  12. [12]

    P. Erdős and L. Pősa: On independent circuits contained in a graph, Canad. J. Math. 17 (1965), 347–352.

    MathSciNet  Article  Google Scholar 

  13. [13]

    B. Estellon: Algorithmes de couverture et d’augmentation de graphes sous contraintes de distance, Thèse de doctorat, Université de Marseille, 2007.

  14. [14]

    S. Fiorini, N. Hardy, B. Reed and A. Vetta: Approximate min-max relations for odd cycles in planar graphs, Math. Program. Ser. B 110 (2007), 71–91.

    MathSciNet  Article  Google Scholar 

  15. [15]

    J. Fox and J. Pach: Touching strings, manuscript, 2012.

  16. [16]

    M. X. Goemans and D. P. Williamson: Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs, Combinatorial 17 (1997), 1–23.

    Article  Google Scholar 

  17. [17]

    D. Haussler and E. Welzl: ϵ-nets and simplex range queries, Discrete Comput. Geom. 2 (1987), 127–151.

    MathSciNet  Article  Google Scholar 

  18. [18]

    A. Kostochka: Lower bound of the Hadwiger number of graphs by their average degree, Combinatorica 4 (1984), 307–316.

    MathSciNet  Article  Google Scholar 

  19. [19]

    R. Krauthgamer and J. Lee: The intrinsic dimensionality of graphs, Combinatorica 27 (2003), 438–447.

    MathSciNet  MATH  Google Scholar 

  20. [20]

    J. Li and M. Parter: Planar diameter via metric compression, in: Proc. 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC), 152–163, 2019.

  21. [21]

    J. Matoušek: Approximations and Optimal Geometric Divide-and-Conquer, J. Comput. System Sci. 50 (1995), 203–208.

    MathSciNet  Article  Google Scholar 

  22. [22]

    B. Reed and F. B. Shepherd: The Gallai-Younger conjecture for planar graphs, Combinatorica 16 (1996), 555–566.

    MathSciNet  Article  Google Scholar 

  23. [23]

    B. Reed, N. Robertson, P. Seymour and R. Thomas: Packing directed circuits, Combinatorica 16 (1996), 535–554.

    MathSciNet  Article  Google Scholar 

  24. [24]

    A. Thomason: An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984), 261–265.

    MathSciNet  Article  Google Scholar 

  25. [25]

    P. Turán: On an extremal problem in graph theory, Mat. és Fiz. Lapok 48 (1941), 436–452 (in Hungarian).

    MathSciNet  Google Scholar 

  26. [26]

    V. Vapnik and A. Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities, Theor. Prob. Appl. 16 (1971), 264–280.

    Article  Google Scholar 

  27. [27]

    V. Wei: A lower bound on the stability number of a simple graph, Technical report, Bell Labs, 1981.

  28. [28]

    D. Wood: Cliques in graphs excluding a complete graph minor, Electron. J. Combin. 23 (2016), P3.18.

    MathSciNet  Article  Google Scholar 

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We thank the two anonymous reviewers for their detailed comments and suggestions.

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Corresponding author

Correspondence to Louis Esperet.

Additional information

N. Bousquet is supported by ANR Project DISTANCIA (ANR-17-CE40-0015). W. Cames van Batenburg, G. Joret, and F. Pirot are supported by an ARC grant from the Wallonia-Brussels Federation of Belgium. L. Esperet is supported by ANR Projects GATO (ANR-16-CE40-0009-01) and GrR (ANR-18-CE40-0032). C. Muller is supported by the Luxembourg National Research Fund (FNR) (Grant Agreement Nr 11628910).

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Bousquet, N., Cames Van Batenburg, W., Esperet, L. et al. Packing and Covering Balls in Graphs Excluding a Minor. Combinatorica 41, 299–318 (2021).

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Mathematics Subject Classification (2010)

  • 05C10
  • 05C12
  • 05C65
  • 05C69
  • 05C83