A new characterization of P 4-connected graphs

Abstract

A graph is said to be P 4-connected if for every partition of its vertices into two nonempty disjoint sets, some P 4 in the graph contains vertices from both sets in the partition. A P 4-chain is a sequence of vertices such that every four consecutive ones induce a P 4. The main result of this work states that a graph is P 4-connected if and only if each pair of vertices is connected by a P 4chain. Our proof relies, in part, on a linear-time algorithm that, given two distinct vertices, exhibits a P 4-chain connecting them. In addition to shedding new light on the structure of P 4-connected graphs, our result extends a previously known theorem about the P 4-structure of unbreakable graphs.