A proof of a conjecture on diameter 2-critical graphs whose complements are claw-free

Abstract

A graph $G$ is diameter 2-critical if its diameter is 2, and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter 2-critical graph of order $n$ is at most $n^2/4$ and that the extremal graphs are complete bipartite graphs with equal size partite sets. We use an important association with total domination to prove the conjecture for the graphs whose complements are claw-free.