Packing and Covering Balls in Graphs Excluding a Minor

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.