A linear time algorithm for constructing tree 3-spanner in simple chordal bipartite graphs

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. Given a graph G and an integer t, the tree t-spanner problem asks whether G admits a tree t-spanner. It is known that the tree t-spanner problem is NP-complete for chordal bipartite graphs for t ges 5 whereas the complexity status of the cases t = 3 and t = 4 are open. In this paper, we study the tree 3- spanner problem in simple chordal bipartite graphs which is a subclass of chordal bipartite graphs. We have shown that this class need not admit tree 3-spanner in general. First, we present a structural characterization of tree 3- spanner admissible simple chordal bipartite graphs. Based on this characterization, we propose a linear time algorithm to recognize tree 3-spanner admissible simple chordal bipartite graphs. Finally, we present a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible simple chordal bipartite graph.