A Meyniel obstruction is an odd cycle with at least five vertices and at most one chord. A graph is Meyniel if and only if it has no Meyniel obstruction as an induced subgraph. Here we give a O(n2) ...algorithm that, for any graph, finds either a clique and a coloring of the same size or a Meyniel obstruction. We also give a O(n3) algorithm that, for any graph, finds either a strong stable set recognizable in polynomial time or a Meyniel obstruction.
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In this paper we present a 2-approximation algorithm for the k -center problem with triangle inequality. This result is "best possible" since for any < 2 the existence of -approximation algorithm ...would imply that P = NP . It should be noted that no -approximation algorithm, for any constant , has been reported to date. Linear programming duality theory provides interesting insight to the problem and enables us to derive, in O (| E | log | E |) time, a solution with value no more than twice the k -center optimal value.
A by-product of the analysis is an O (| E |) algorithm that identifies a dominating set in G 2 , the square of a graph G , the size of which is no larger than the size of the minimum dominating set in the graph G . The key combinatorial object used is called a strong stable set, and we prove the NP -completeness of the corresponding decision problem.
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