RECOMB '01 Proceedings of the fifth annual international conference on Computational biology
Coloring Algorithms for Tolerance Graphs: Reasoning and Scheduling with Interval Constraints
AISC '02/Calculemus '02 Proceedings of the Joint International Conferences on Artificial Intelligence, Automated Reasoning, and Symbolic Computation
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A graph G = (V.E) is said to be weakly triangulated if neither G not G, the complement of G, contain chordless or induced cycles of length greater than four. Ryan Hayward showed that weakly triangulated graphs are perfect. Later, Hayward, Huang and Maffray obtained ) (e.v.) algorithms to find a maximum clique and a minimum coloring of a weakly triangulated graph. Performing these algorithms on the complement graph gives O (v) algorithms to find a maximum independent set and a minimum clique cover of such a graph. It was shown in [13-16] that weakly triangulated graphs play a crucial role in polygon decomposition problems. Several polygon decomposition problems can be formulated as the problem of covering a weakly triangulated graph with a minimum number of cliques. Motivated by this, we now improve on the algorithms of Hayward, Hoang and Maffray by providing O (e.v) algorithms to find a maximum clique and a minimum coloring of a weakly triangulated graph. We thus obtain an O (v) algorithm to find a maximum independent set and a minimum clique cover of such a graph. We also provide O (v) algorithms for weighted versions of these problems.