Tree-like P4-connected graphs Original Research Article

A graph is P4-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 P4-connected if and only if any two vertices are connected by a P4-chain, that is, a sequence of vertices such that every four consecutive ones induce a P4. In this paper we study graphs where each induced subgraph contains a vertex which belongs to at most one P4. It turns out that the P4-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 P4s, and by the uniqueness of P4-chains connecting two vertices.