Tree Spanners on Chordal Graphs: Complexity, Algorithms, Open Problems

A tree t-spanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The Tree t-Spanner problem asks whether a graph admits a tree t-spanner, given t. We substantially strengthen the hardness result of Cai and Corneil [SIAM J. Discrete Math. 8 (1995) 359– 387] by showing that, for any t ≥ 4, Tree t-Spanner is NP-complete even on chordal graphs of diameter at most t+1 (if t is even), respectively, at most t + 2 (if t is odd). Then we point out that every chordal graph of diameter at most t − 1 (respectively, t − 2) admits a tree t-spanner whenever t ≥ 2 is even (respectively, t ≥ 3 is odd), and such a tree spanner can be constructed in linear time. The complexity status of Tree 3-Spanner still remains open for chordal graphs, even on the subclass of undirected path graphs that are strongly chordal as well. For other important subclasses of chordal graphs, such as very strongly chordal graphs (containing all interval graphs), 1-split graphs (containing all split graphs) and chordal graphs of diameter at most 2, we are able to decide Tree 3-Spanner efficiently.