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 VC-dimension 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 Vax\`es asserting that every planar graph of diameter $2\ell$ can be covered by a bounded number of balls of radius $\ell$. In fact, they obtained the existence of a function $f$ such that every set $\cal F$ of balls of radius $\ell$ in a planar graph admits a hitting set of size $f(\nu)$ where $\nu$ is the maximum number of pairwise disjoint elements of $\cal F$. Our goal is to generalize the proof of Chepoi, Estellon and Vax\`es 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 VC-dimension has the Erd\H{o}s-P\'osa property.