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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  • 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.”

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Acknowledgement

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

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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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Mathematics Subject Classification (2010)

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