On Murty-Simon Conjecture

On Murty-Simon Conjecture II

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 ⌊ n2 4 ⌋ and the extremal graph is the complete bipartite graph Kb 2 c,d 2 e. In the series papers [7–9], the Murty-Simon Conjecture stated by Haynes et al. is not the...

Progress on the Murty-Simon Conjecture on diameter-2 critical graphs: a survey

A graph G is diameter 2-critical if its diameter is two and the deletion of any edgeincreases the diameter. Murty and Simon conjectured that the number of edges in adiameter-2-critical graph G of order n is at most bn/4c and that the extremal graphsare the complete bipartite graphs Kbn/2c,dn/2e. We survey the progress made to dateon this conjecture, concentrating mainly on recen...

On a conjecture of Murty and Simon on diameter two critical graphs II

A characterization of diameter-2-critical graphs with no antihole of length four

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 n2/4 ...

Maximal and Minimal Vertex - critical Graphsof Diameter

A graph is vertex-critical (edge-critical) if deleting any vertex (edge) increases its diameter. A conjecture of Simon and Murty stated thatèvery edge-critical graph of diameter two on vertices contains at most 1 4 2 edges'. This conjecture has been established for suuciently large. For vertex-critical graphs, little is known about the number of edges. Plesn ik implicitly asked whether it is al...