VC-dimension and Erdős–Pósa property
Résumé
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 whose 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 et al. (2007) asserting that every planar graph of diameter 2ℓ can be covered by a bounded number of balls of radius ℓ. In fact, they obtained the existence of a function f such that every set F of balls of radius ℓ 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 Chepoi et al. (2007) 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 Erdős–Pósa property.