A graph is diameter two edge-critical if its diameter is two and the deletion
of any edge increases the diameter. Murty and Simon conjectured that the number
of edges in a diameter two edge-critical graph on n vertices is at most
β4n2ββ and the extremal graph is the complete
bipartite graph Kβ2nββ,β2nβββ.
In the series papers [7-9], the Murty-Simon Conjecture stated by Haynes et al.
is not the original conjecture, indeed, it is only for the diameter two
edge-critical graphs of even order. In this paper, we completely prove the
Murty-Simon Conjecture for the graphs whose complements have vertex
connectivity β, where β=1,2,3; and for the graphs whose
complements have an independent vertex cut of cardinality at least three.Comment: 9 pages, submitted for publication on May 10, 201