Tree 3-spanners of diameter at most 5

A subgraph T of a graph G is a tree t-spanner of G if and only if T is a tree and for every pair of vertices in G their distance in T is at most t times their distance in G, where t is called a stretch factor of T . An efficient algorithm to determine whether a graph admits a tree 2-spanner has been developed, while for each t ≥ 4 the problem to determine whether a graph admits a tree t-spanner has been proven to be NP-complete. Although it is not known whether it is tractable to decide graphs that admit a tree 3-spanner, an efficient algorithm to determine whether a graph admits a tree 3-spanner of diameter at most 5 is presented.