The clique operator on circular-arc graphs

Abstract

a b s t r a c tA circular-arc graph G is the intersection graph of a collection of arcs on the circle and such a collection is called a model of G. Say that the model is proper when no arc of the collection contains another one, it is Helly when the arcs satisfy the Helly Property, while the model is proper Helly when it is simultaneously proper and Helly. A graph admitting a Helly (resp. proper Helly) model is called a Helly (resp. proper Helly) circular-arc graph. The clique graph K (G) of a graph G is the intersection graph of its cliques. The iterated clique graph K i (G) of G is defined by K 0 (G) = G and K i+1 (G) = K (K i (G)). In this paper, we consider two problems on clique graphs of circular-arc graphs. The first is to characterize clique graphs of Helly circular-arc graphs and proper Helly circular-arc graphs. The second is to characterize the graph to which a general circular-arc graph K -converges, if it is K -convergent. We propose complete solutions to both problems, extending the partial results known so far. The methods lead to linear time recognition algorithms, for both problems.

On the termination of some biclique operators on multipartite graphs

“…For the well-known clique graph operator (see [24] for a survey) the question of convergence has received a lot of attention [23,6]. Most of the efforts focussed on obtaining convergence results, or divergence results, for some particular graphs or graph classes [17,16,18,20]. Similar questions have been addressed recently for the biclique graph operator [12,13], which also operates on graphs 4 but using bicliques instead of cliques.…”

On the clique behavior of graphs of low degree

“…The problem of determining the behavior of G under iterated applications of the clique operator is one of the main topics in this theory, as the clique graph operator is considered one of the most complex graph operators ( [17]). There are many families of graphs for which criterions have been proved, and in some cases the behavior can even be determined in polynomial time (see for example [7], [8], [11], [9], [10], [13], [14], [6], [4], [1]), however the problem has been found to be undecidable for automatic graphs ( [2]).…”

Forbidden induced subgraphs of normal Helly circular-arc graphs: Characterization and detection

“…This is tight as any odd hole, which has ω = 2 and needs at least three colors, is a normal Helly circular-arc graph. In the study of convergence of circular-arc graphs under the clique operator, Lin et al [9] observed that normal Helly circular-arc graphs arose naturally. They then [10] undertook a systematic study of normal Helly circular-arc graphs as well as its subclass.…”