# 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*_{4}chain. 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.

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