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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)
##### Keywords:
Helly circular-arc graphs
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##### References:
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