How to decrease the diameter of triangle-free graphs

Assume that G is a triangle-free graph. Let h(d)(G) be the minimum number of edges one has to add to G to get a graph of diameter at most d which is still triangle-free. It is shown that h(2)(G)=Theta(n log n) for connected graphs of order n and of fixed maximum degree. The proof is based on relations of h(2)(G) and the clique-cover number of edges of graphs. It is also shown that the maximum value of h(2)(G) over (triangle-free) graphs of order n is inverted right perpendicular n/2 - 1 inverted left perpendicular [n/2 - 1]. The behavior of h(3)(G) is different, its maximum value is n-1. We could not decide whether h(4)(G) less than or equal to (1- epsilon)n for connected (triangle-free) graphs of order n with a positive epsilon.