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 also true that 1 4 2 is an upper bound for the number of edges in a vertex-critical graph of diameter two on vertices. In this paper, we construct , for each 5 except = 6, a vertex-critical graph of diameter two on vertices with at least 1 2 2 ? p 2 3 2 + c 1 edges, where c 1 is some constant. We show that each vertex-critical graph contains at most 1 2 2 ? p 2 2 3 2 + c 2 edges, where c 2 is some constant. In addition, we also construct, for each 5 except = 6, a vertex-critical graph of diameter two on vertices with at most 1 2 (5 ? 12) edges. We show each vertex-critical graph must contain at least 1 2 (5 ? 29) edges. This second result is comparable to a result of Murty on the minimum number of edges possible in an edge-critical graph of diameter two.