Tree spanners of small diameter

A graph that contains a spanning tree of diameter at most t clearly admits a tree t-spanner, since a tree t-spanner of a graph G is a sub tree of G such that the distance between pairs of vertices in the tree is at most t times their distance in G. In this paper, graphs that admit a tree t-spanner of diameter at most t + 1 are studied. For t equal to 1 or 2 the problem has been solved. For t = 3 we present an algorithm that determines if a graph admits a tree 3-spanner of diameter at most 4. For t ≥ 4 it is proved that it is an NP-complete problem to decide whether a graph admits a tree t-spanner of diameter at most t+ 1.