Nonexistence of triples of nonisomorphic connected graphs with isomorphic connected $P_3$-graphs

TL;DR

This paper proves that no three mutually nonisomorphic connected graphs can have isomorphic connected $P_3$-graphs, resolving a long-standing open problem in graph theory.

Contribution

It provides a definitive proof that such triples do not exist, settling a problem posed in 1989.

Findings

No triple of mutually nonisomorphic connected graphs has isomorphic connected $P_3$-graphs.

The problem posed by Broersma and Hoede is completely solved.

The result advances understanding of graph isomorphism and path graph relationships.

Abstract

In the paper "Broersma and Hoede, {\it Path graphs}, J. Graph Theory {\bf 13} (1989) 427-444", the authors proposed a problem whether there is a triple of mutually nonisomorphic connected graphs which have an isomorphic connected $P_3$-graph. For a long time, this problem remains unanswered. In this paper, we give it a negative answer that there is no such triple, and thus completely solve this problem.