A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. 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. As a consequence of our characterization, we prove the Murty-Simon Conjecture for graphs with no antihole of length four.
 Fan G., On diameter 2-critical graphs, Discrete Math., 1987, 67(3), 235–240 http://dx.doi.org/10.1016/0012-365X(87)90174-9
 Füredi Z., The maximum number of edges in a minimal graph of diameter 2, J. Graph Theory, 1992, 16(1), 81–98 http://dx.doi.org/10.1002/jgt.3190160110
 Hanson D., Wang P., A note on extremal total domination edge critical graphs, Util. Math., 2003, 63, 89–96
 Haynes T.W., Hedetniemi S.T., Slater P.J., Fundamentals of Domination in Graphs, Monogr. Textbooks Pure Appl. Math., 208, Marcel Dekker, New York, 1998
 Haynes T.W., Henning M.A., van der Merwe L.C., Yeo A., On a conjecture of Murty and Simon on diameter 2-critical graphs, Discrete Math., 2011, 311(17), 1918–1924 http://dx.doi.org/10.1016/j.disc.2011.05.007
 Haynes T.W., Henning M.A., Yeo A., A proof of a conjecture on diameter 2-critical graphs whose complements are claw-free, Discrete Optim., 2011, 8(3), 495–501 http://dx.doi.org/10.1016/j.disopt.2011.04.003
 Henning M.A., A survey of selected recent results on total domination in graphs, Discrete Math., 2009, 309(1), 32–63 http://dx.doi.org/10.1016/j.disc.2007.12.044
 van der Merwe L.C., Total Domination Critical Graphs, PhD thesis, University of South Africa, 1998
 van der Merwe L.C., Mynhardt C.M., Haynes T.W., Total domination edge critical graphs, Util. Math., 1998, 54, 229–240
 Murty U.S.R., On critical graphs of diameter 2, Math. Mag., 1968, 41, 138–140 http://dx.doi.org/10.2307/2688184
 Plesník J., Critical graphs of given diameter, Acta Fac. Rerum Natur. Univ. Comenian. Math., 1975, 30, 71–93 (in Slovak)
 Xu J.M., Proof of a conjecture of Simon and Murty, J. Math. Res. Exposition, 1984, 4, 85–86 (in Chinese)
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