Tree 3-spanners in 2-sep chordal graphs: Characterization and algorithms

A spanning tree T of a graph G is said to be 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 <= 2. The case t = 3 still remains open. A chordal graph is called a 2-sep chordal graph if all of its minimal a - b vertex separators for every pair of non-adjacent vertices a and b are of size two. It is known that not all 2-sep chordal graphs admit tree 3-spanners This paper presents a structural characterization and a linear time recognition algorithm of tree 3-spanner admissible 2-sep chordal graphs. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep chordal graph is proposed. (C) 2010 Elsevier B.V. All rights reserved.