The clique operator on cographs and serial graphs

Larrión, F.; Mello, Célia Picinin de; Morgana, A.; Neumann-Lara, Víctor; Pizaña, Miguel A.

doi:10.1016/j.disc.2003.10.023

Cited by 25 publications

(21 citation statements)

References 24 publications

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“…It is known that every Helly graph is clique convergent [7], but the opposite is not true in general [7,4,13,9]. However, there are several families of graphs which are known to be divergent precisely when they are not Helly such as cographs [10], complements of cycles [20,15], powers of cycles [15] and chessboard graphs [16]. We shall prove that the circulants studied here also exhibit this property.…”

mentioning

confidence: 67%

On the clique behavior of circulants with three small jumps

Larrión

Pizaña

Villarroel-Flores

2009

Electronic Notes in Discrete Mathematics

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confidence: 67%

On the clique behavior of circulants with three small jumps

Larrión

Pizaña

Villarroel-Flores

2009

Electronic Notes in Discrete Mathematics

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“…Nevertheless, polynomial-time algorithms to decide the K -behavior of a few classes are known. This is the case for cographs [6], P 4 -tidy graphs [7] and complete multipartite graphs [8]. Clique-Helly graphs K -converge to graphs with period either 1 or 2 [9], interval graphs are K -null and octahedra of dimension at least 3 K -diverge.…”

Section: Introductionmentioning

confidence: 99%

The clique operator on circular-arc graphs

Lin

Soulignac

Szwarcfiter

2010

Discrete Applied Mathematics

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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.

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“…For the clique-Helly graph class, graphs which are convergent to the trivial graph have been characterized in [3]. Cographs, P 4 -tidy graphs, and circulararc graphs are examples of classes where the different behaviors were also characterized [7,22]. On the other hand, divergent graphs were considered.…”

Section: Introductionmentioning

confidence: 99%

On the Iterated Edge-Biclique Operator

Legay¹,

Montero

2019

Electronic Notes in Theoretical Computer Science

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A biclique of a graph G is a maximal induced complete bipartite subgraph of G. The edge-biclique graph of G, KB e (G), is the edge-intersection graph of the bicliques of G. A graph G diverges (resp. converges or is periodic) under an operator H whenever lim k→∞ |V (H k (G))| = ∞ (resp. lim k→∞ H k (G) = H m (G) for some m or H k (G) = H k+s (G) for some k and s ≥ 2). The iterated edge-biclique graph of G, KB k e (G), is the graph obtained by applying the edge-biclique operator k successive times to G. In this paper, we first study the connectivity relation between G and KB e (G). Next, we study the iterated edge-biclique operator KB e . In particular, we give sufficient conditions for a graph to be convergent or divergent under the operator KB e , we characterize the behavior of burgeon graphs and we propose some general conjectures on the subject.

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The clique operator on cographs and serial graphs

Cited by 25 publications

References 24 publications

On the clique behavior of circulants with three small jumps

On the clique behavior of circulants with three small jumps

The clique operator on circular-arc graphs

On the Iterated Edge-Biclique Operator

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