A characterization of P5P5-free, diameter-2-critical graphs

A graph GG is diameter-22-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-22-critical graphs with no induced path on five vertices. Murty and Simon conjectured that the number of edges in a diameter-22-critical graph of order nn is at most ⌊n2/4⌋⌊n2/4⌋ and that the extremal graphs are the complete bipartite graphs with partite sets whose cardinality differs by at most one. We use an association with total domination to prove that if GG is a diameter-22-critical graph with no induced path P5P5, then GG is triangle-free. As a consequence, we observe that the Murty–Simon Conjecture is true for P5P5-free, diameter-2-critical graphs by Turán’s Theorem. Finally we characterize these graphs by characterizing their complements.