  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
$\lfloor \frac{n^{2}}{4} \rfloor$ and the extremal graph is the complete
bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$.
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 $\ell$, where $\ell = 1, 2, 3$; and for the graphs whose
complements have an independent vertex cut of cardinality at least three.
