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Packing and Covering Balls in Graphs Excluding a Minor

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Abstract

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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Change history

  • 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.”

References

  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.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

  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.

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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Acknowledgement

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

Author information

Authors and Affiliations

  1. Laboratoire G-SCOP, CNRS Univ. Grenoble Alpes, Grenoble, France

    Nicolas Bousquet, Louis Esperet & François Pirot

  2. Algorithms Research Group, University of Bergen, Bergen, Norway

    William Lochet

  3. Département d’Informatique, Université Libre de Bruxelles, Brussels, Belgium

    Wouter Cames Van Batenburg & Gwenaël Joret

  4. Département de Mathématique, Université Libre de Bruxelles, Brussels, Belgium

    Carole Muller

  5. Département d’Informatique, Université Libre de Bruxelles, Brussels, Belgium

    François Pirot

Authors
  1. Nicolas Bousquet
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  2. Wouter Cames Van Batenburg
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  3. Louis Esperet
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  4. Gwenaël Joret
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  5. William Lochet
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  6. Carole Muller
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  7. François Pirot
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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). https://doi.org/10.1007/s00493-020-4423-3

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  • DOI: https://doi.org/10.1007/s00493-020-4423-3

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