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Algorithmic aspects of clique-transversal and clique-independent sets. (English) Zbl 0948.68135
Summary: A Minimum Clique-Transversal set MCT\((G)\) of a graph \(G= (V, E)\) is a set \(S\subseteq V\) of minimum cardinality that meets all maximal cliques in \(G\). A Maximum Clique-Independent set MCI\((G)\) of G is a set of maximum number of pairwise vertex-disjoint maximal cliques. We prove that the problem of finding an MCT\((G)\) and an MCI\((G)\) is NP-hard for cocomparability, planar, line and total graphs. As an interesting corollary we obtain that the problem of finding a minimum number of elements of a poset to meet all maximal antichains of the poset remains NP-hard even if the poset has height two, thereby generalizing a result of D. Duffus, H. A. Kierstead and W. T. Trotter [J. Comb. Theory, Ser. A 58, No. 1, 158-164 (1991; Zbl 0757.06001)]. We present a polynomial algorithm for the above problems on Helly circular-arc graphs which is the first such algorithm for a class of graphs that is not clique-perfect. We also present polynomial algorithms for the weighted version of the clique-transversal problem on strongly chordal graphs, chordal graphs of bounded clique size, and cographs. The algorithms presented run in linear time for strongly chordal graphs and cographs. These mark the first attempts at the weighted version of the problem.

MSC:
68R10 Graph theory (including graph drawing) in computer science
68Q25 Analysis of algorithms and problem complexity
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C85 Graph algorithms (graph-theoretic aspects)
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