Tree-like P4-connected graphs

Abstract

A graph is P 4-connected if for every partition of its vertices into two nonempty disjoint sets there is a chordless path on four vertices which contains vertices from both sets in the partition. An alternative characterization states that a graph is P 4-connected if and only if any two vertices are connected by a P 4-chain, that is, a sequence of vertices such that every four consecutive ones induce a P 4. In this paper we study graphs where each induced subgraph contains a vertex which belongs to at most one P 4. It turns out that the P 4-connected components of these graphs are provided with structural properties which can be expressed in a quite analogous way to the numerous characterizations of ordinary trees. Among others, we present characterizations by forbidden subgraphs, in terms of the number of P 4s, and by the uniqueness of P 4-chains connecting two vertices.