Tree 3-spanners in 2-sep directed path graphs: Characterization, recognition, and construction

Abstract

A spanning tree T of a graph G is called a tree t - spanner, if the distance between any two vertices in T is at most t -times their distance in G . A graph that has a tree t -spanner is called a tree t - spanner admissible graph. The problem of deciding whether a graph is tree t -spanner admissible is NP-complete for any fixed t ≥ 4 , and is linearly solvable for t = 1 and t = 2 . The case t = 3 still remains open. A directed path graph is called a 2-sep directed path graph if all of its minimal a − b vertex separator for every pair of non-adjacent vertices a and b are of size two. Le and Le [H.-O. Le, V.B. Le, Optimal tree 3-spanners in directed path graphs, Networks 34 (2) (1999) 81–87] showed that directed path graphs admit tree 3-spanners. However, this result has been shown to be incorrect by Panda and Das [B.S. Panda, Anita Das, On tree 3-spanners in directed path graphs, Networks 50 (3) (2007) 203–210]. In fact, this paper observes that even the class of 2-sep directed path graphs, which is a proper subclass of directed path graphs, need not admit tree 3-spanners in general. It, then, presents a structural characterization of tree 3-spanner admissible 2-sep directed path graphs. Based on this characterization, a linear time recognition algorithm for tree 3-spanner admissible 2-sep directed path graphs is presented. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep directed path graph is proposed.