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Algorithms for clique-independent sets on subclasses of circular-arc graphs. (English) Zbl 1104.05054
Authors’ abstract: A circular arc (CA) graph is the intersection graph of arcs on a circle. A Helly circular-arc (HCA) graph is a CA graph admitting a model whose arcs satisfy the Helly property. A clique-independent set of a graph is a set of pairwise disjoint cliques of the graph. It is NP-hard to compute the maximum cardinality of a clique-independent set for a general graph.
In the present paper, we propose polynomial time algorithms for finding the maximum cardinality and weight of a clique-independent set of a \(\overline{3K_2}\)-free CA graph. Also, we apply the algorithms to the special case of an HCA graph. The complexity of the proposed algorithm for the cardinality problem in HCA graphs is \(O(n)\). This represents an improvement over the existing algorithm by V. Guruswami and C. Pandu Rangan [Algorithmic aspects of clique-transversal and clique-independent sets, Discrete Appl. Math. 100, No. 3, 183–202 (2000; Zbl 0948.68135)], whose complexity is \(O(n^2)\). These algorithms suppose that an HCA model of the graph is given.

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