Some exact values for p(t, d)

Let G = (V, E) be a graph When a graph is used to model the linkage structure of communication networks, the diameter of a graph gives the length of longest path among all the shortest paths between any two vertices of the graph, F(t, d) denoted the minim um diameter of an altered graph obtained by adding t - extra edges to a graph with diameter d. Let P(t, d) denoted the minimum diameter of a graph obtained by adding t - extra edges to a path with diameter d. Knowing that F(t, d) = p(t, d). In this paper we prove that p(t, 3t) = 4, for t >= 4, p(t, d) = 4 where (t >= 4 and 3t +1 <= d <= 3t + 3) and show a construction to p(t, d) <= k + 3 where (3 <= t <= 5) and d=k(t+1)+8.