VC-dimension and Erd\H{o}s-P\'osa property

Let G = (V,E) be a graph. A k-neighborhood in G is a set of vertices consisting of all the vertices at distance at most k from some vertex of G. The hypergraph on vertex set V which edge set consists of all the k-neighborhoods of G for all k is the neighborhood hypergraph of G. Our goal in this paper is to investigate the complexity of a graph in terms of its neighborhoods. Precisely, we define the distance VCdimension of a graph G as the maximum taken over all induced subgraphs G′ of G of the VC-dimension of the neighborhood hypergraph of G′. For a class of graphs, having bounded distance VC-dimension both generalizes minor closed classes and graphs with bounded clique-width. Our motivation is a result of Chepoi, Estellon and Vaxès [5] asserting that every planar graph of diameter 2` can be covered by a bounded number of balls of radius `. In fact, they obtained the existence of a function f such that every set F of balls of radius ` in a planar graph admits a hitting set of size f(ν) where ν is the maximum number of pairwise disjoint elements of F . Our goal is to generalize the proof of [5] with the unique assumption of bounded distance VC-dimension of neighborhoods. In other words, the set of balls of fixed radius in a graph with bounded distance VCdimension has the Erdős-Pósa property.