, Since free(f ) = 1, by Lemma 12(d), |S u | ? |S v | ? 1. As in the proof of Lemma 12, we let free(P uv , f ) be the number of f-free non-edges incident with a vertex of P uv, Case 2: P uv is nonempty

, is incident with a vertex p of P uv , say it is pt for some vertex t ? S v. Consider the sets S u (p) and S v (p) and the function m as defined in Lemma 12, By Lemma, vol.11

, ? 2 edges, and we are done. Thus, we assume that ||X| ? |Y || ? 1, Recall that by Lemma, vol.12

|. |x|-=-|s-u-|-+-|s-uv-|-+-|p-uv-|-+-1-?-|s-v-|-+-|s-uv-|-+-|p-uv-|-=-|y-|-?-1-+-|s-uv-|-+-|p-uv,

|. Since and . |y, it follows that |S uv | + |P uv | ? {1, 2}, and thus either |S uv | = |P uv | = 1 or S uv = ? and |P uv | ? {1

=. Suppose-that-|s-uv-|-=-|p-uv-|-=-1.-then-p-uv, . {p}, and . |s-u-|-=-|s-v-|?1, Observe that, if vt is critical for the pair {s, t} with S uv = {s}, then s has no neighbour in S v (since t is adjacent to every other vertex in S v ), and thus all but one of the non-edges between s and S v are f-free (their only possible preimage by f is sv), The latter implies S u = S u (p), and again (as in the case free

. Then-p-uv-=-{p} and . |s-u-|-=-|s-v-|, Assume first that there exists a vertex m(y) ? S u (p) (for a certain y ? S v (p)) such that m(y)t / ? E. Then the non-edge m(y)t must have a preimage by the function f , and there are two possibilities. First, if either f (yt) = m(y)t or f (m(x)m(y)) = m(y)t, then by Lemma 12(c) the non-edge m(x)y is f-free, a contradiction, However, by Lemma 8(c), w has a neighbour in S v. Thus, w and the vertex in S u (p) have two common neighbours

, Hence, we have shown that the f-free non-edge is not incident with p

, S uv. Then, S uv = ? and free(P uv , f ) = 0. Thus, by Lemma 12(d), we have |S u | ? |S v |. But then, ||X| ? |Y || = (|S u | + |P uv | + |S uv |) ? |S v | ? 2, and by Lemma, vol.11

, Then, since free(P uv , f ) = 0, by Lemma 12(d), we deduce that S v = S v (p), and |S u | ? |S v |, Graphs and Combinatorics, vol.12, issue.5, pp.1163-1176, 2015.

J. Balogh and . Pet?í?ková, The number of the maximal triangle-free graphs, Bulletin of the London Mathematical society, vol.46, issue.5, pp.1003-1006, 2014.

C. Barefoot, K. Casey, D. C. Fisher, K. Fraughnaugh, and F. Harary, Size in maximal triangle-free graphs and minimal graphs of diameter 2, Discrete Mathematics, vol.138, issue.1-3, pp.93-99, 1995.

B. Bollobás and S. Eldridge, On graphs with diameter 2, Journal of Combinatorial Theory, Series B, vol.22, pp.201-205, 1976.

L. Caccetta and R. Häggkvist, On diameter critical graphs, Discrete Mathematics, vol.28, issue.3, pp.223-229, 1979.

R. B. Eggleton and J. A. Macdougall, Minimally triangle-saturated graphs, adjoining a single vertex, Australasian Journal of Combinatorics, vol.25, pp.263-278, 2002.

P. Erd?-os, A. Rényi, and V. T. Sós, On a problem of graph theory, Studia Scientiarum Mathematicarum Hungarica, vol.1, pp.215-235, 1966.

G. Fan, On diameter 2-critical graphs, Discrete Mathematics, vol.67, issue.3, pp.235-240, 1987.

Z. Füredi, The maximum number of edges in a minimal graph of diameter 2, Journal of Graph Theory, vol.16, issue.1, pp.81-98, 1992.

W. Goddard and D. J. Kleitman, A note on maximal triangle-free graphs, Journal of Graph Theory, vol.17, issue.5, pp.629-631, 1993.

D. Hanson and P. Wang, A note on extremal total domination edge critical graphs, Utilitas Mathematica, vol.63, pp.89-96, 2003.

T. W. Haynes and M. A. Henning, A characterization of diameter-2-critical graphs whose complements are diamond-free, Discrete Appl. Math, vol.160, 1979.

T. W. Haynes, M. A. Henning, L. C. Van-der-merwe, and A. Yeo, On a conjecture of Murty and Simon on diameter 2-critical graphs, Discrete Mathematics, vol.311, issue.17, pp.1918-1924, 2011.

T. W. Haynes, M. A. Henning, L. C. Van-der-merwe, and A. Yeo, A maximum degree theorem for diameter-2-critical graphs, Central European Journbal of Mathematics, vol.12, issue.12, pp.1882-1889, 2014.

T. W. Haynes, M. A. Henning, L. C. Van-der-merwe, and A. Yeo, Progress on the Murty-Simon conjecture on diameter-2 critical graphs: a survey, Journal of Combinatorial Optimization, vol.30, issue.3, pp.579-595, 2015.

T. W. Haynes, M. A. Henning, L. C. Van-der-merwe, and A. Yeo, A completion of three proofs related to the Murty-Simon conjecture, 2015.

T. W. Haynes, M. A. Henning, and A. Yeo, A proof of a conjecture on diameter 2-critical graphs whose complements are claw-free, Discrete Optimization, vol.8, issue.3, pp.495-501, 2011.

T. W. Haynes, M. A. Henning, and A. Yeo, On a conjecture of Murty and Simon on diameter two critical graphs II, Discrete Mathematics, vol.312, issue.2, pp.315-323, 2012.

T. W. Haynes, C. M. Mynhardt, and L. C. Van-der-merwe, Total domination edge critical graphs, Utilitas Mathematica, vol.54, pp.229-240, 1998.

T. W. Haynes, C. M. Mynhardt, and L. C. Van-der-merwe, Total domination edge critical graphs with maximum diameter, Discussiones Mathematicae Graph Theory, vol.21, issue.2, pp.187-205, 2001.

A. J. Hoffman and R. R. Singleton, On Moore Graphs of diameter two and three, IBM Journal of Researhc and Development, vol.4, pp.497-504, 1960.

P. Loh and J. Ma, Diameter critical graphs, Journal of Combinatorial Theory, Series B, vol.117, pp.34-58, 2016.

J. A. Macdougall and R. B. Eggleton, Triangle-free and triangle-saturated Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, vol.25, pp.3-21, 1997.

W. Mantel, Problem 28, Wiskundige Opgaven, vol.10, pp.60-61, 1906.

O. Ore, Diameters in graphs, Journal of Combinatorial Theory, vol.5, pp.75-81, 1968.

J. Plesník, Critical graphs of given diameter. Acta Facultatis Rerum Naturalium Universitatis Comenianae Mathematica, vol.30, pp.71-93, 1975.

J. Plesník, On minimal graphs of diameter 2 with every edge in a 3-cycle, Mathematica Slovaca, vol.36, issue.2, pp.145-149, 1986.

T. Wang, On Murty-Simon Conjecture, 2012.