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On cliques of Helly circular-arc graphs. (English) Zbl 1341.05196
Liebling, Th. (ed.) et al., The IV Latin-American algorithms, graphs, and optimization symposium, Puerto Varas, Chile, November 25–29, 2007. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 30, 117-122 (2008).
Summary: A circular-arc graph is the intersection graph of a set of arcs on the circle. It is a Helly circular-arc graph if it has a Helly model, where every maximal clique is the set of arcs that traverse some clique point on the circle. A clique model is a Helly model that identifies one clique point for each maximal clique. A Helly circular-arc graph is proper if it has a Helly model where no arc is a subset of another. In this paper, we show that the clique intersection graphs of Helly circular-arc graphs are related to the proper Helly circular-arc graphs. This yields the first polynomial (linear) time recognition algorithm for the clique graphs of Helly circular-arc graphs. Next, we develop an $$O(n)$$ time algorithm to obtain a clique model of Helly model, improving the previous $$O(n^{2})$$ bound. This gives a linear-time algorithm to find a proper Helly model for the clique graph of a Helly circular-arc graph. As an application, we find a maximum weighted clique of a Helly circular-arc graph in linear time.
For the entire collection see [Zbl 1137.05001].

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C85 Graph algorithms (graph-theoretic aspects) 05C35 Extremal problems in graph theory
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