Packing and covering balls in graphs excluding a minor

01/13/2020

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by   Nicolas Bousquet, et al.

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We prove that for every integer t> 1 there exists a constant c_t such that for every K_t-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 c_t 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.