Tree spanners of bounded degree graphs

A tree t-spanner of a graph G is a spanning tree of G such that the distance between pairs of vertices in the tree is at most t times their distance in G. Deciding tree t-spanner admissible graphs has been proved to be tractable for t < 3 and NP-complete for t > 3, while the complexity status of this problem is unresolved when t = 3. For every t > 2 and b > 0, an efficient dynamic programming algorithm to decide tree t-spanner admissibility of graphs with vertex degrees less than b is presented. Only for t = 3, the algorithm remains efficient, when graphs G with degrees less than b log |V (G)| are examined.