Tree spanners on chordal graphs: complexity and algorithms |
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Authors: | Andreas Brandstdt Feodor F Dragan Hong-Oanh Le Van Bang Le |
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Affiliation: | a Institut für Theoretische Informatik, Fachbereich Informatik, Universität Rostock, 18051, Rostock, Germany b Department of Computer Science, Kent State University, Kent, OH 44242, USA |
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Abstract: | 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 T
t-S
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, T
t-S
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 T
3-S
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 T
3-S
efficiently. |
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Keywords: | Tree spanners Spanners Chordal graphs Distance Spanning trees NP-completeness Efficient graph algorithms |
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