Triangulation and clique separator decomposition of claw-free graphs

Authors:
Anne Berry;Annegret Wagler
Affiliations:
LIMOS UMR CNRS 6158, Ensemble Scientifique des Cézeaux, Université Blaise Pascal, Aubière, France;LIMOS UMR CNRS 6158, Ensemble Scientifique des Cézeaux, Université Blaise Pascal, Aubière, France
Venue:
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
Year:
2012

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Abstract

Finding minimal triangulations of graphs is a well-studied problem with many applications, for instance as first step for efficiently computing graph decompositions in terms of clique separators. Computing a minimal triangulation can be done in O(nm) time and much effort has been invested to improve this time bound for general and special graphs. We propose a recursive algorithm which works for general graphs and runs in linear time if the input is a claw-free graph and the length of its longest path is bounded by a fixed value k. More precisely, our algorithm runs in O(f+km) time if the input is a claw-free graph, where f is the number of fill edges added, and k is the height of the execution tree; we find all the clique minimal separators of the input graph at the same time. Our algorithm can be modified to a robust algorithm which runs within the same time bound: given a non-claw free input, it either triangulates the graph or reports a claw.