The algorithmic complexity of the minus clique-transversal problem

Xu, Guangjun; Shan, Erfang; Kang, Liying; Cheng, T.C.E.

doi:10.1016/j.amc.2006.12.027

Cited by 11 publications

(8 citation statements)

References 14 publications

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“…The clique transversal and clique independent set problems have been widely studied in Andreae et al (1991), Andreae and Flotow (1996), Andreae (1998), Balachandran et al (1996), Bandelt and Mulder (1986), Bonomo et al (2006a), Brandstädt et al (1997), Chang et al (1993Chang et al ( , 1996Chang et al ( , 2001, Dahlhaus et al (1998), Durán et al (2002Durán et al ( , 2006Durán et al ( , 2008, Erdös et al (1992), Guruswami and Rangan (2000), Lee and Chang (2006), Shan et al (2008), Xu et al (2007). Both the clique transversal and clique independent set problems are NP-hard for cocomparability graphs, planar graphs, line graphs, total graphs (Guruswami and Rangan 2000), split graphs, undirected path graphs, and k-trees with unbounded k (Chang et al 1993(Chang et al , 1996, while they are polynomial-time solvable for balanced graphs (Bonomo et al 2006a;Dahlhaus et al 1998), comparability graphs (Balachandran et al 1996), distance-hereditary graphs (Lee and Chang 2006), Helly circular-arc graphs (Guruswami and Rangan 2000;Durán et al 2008), doubly chordal graphs (Brandstädt et al 1997), and strongly chordal graphs (Brandstädt et al 1997;Chang et al 1993Chang et al , 1996.…”

Section: For Any Vertexmentioning

confidence: 98%

“…Both the clique transversal and clique independent set problems are NP-hard for cocomparability graphs, planar graphs, line graphs, total graphs (Guruswami and Rangan 2000), split graphs, undirected path graphs, and k-trees with unbounded k (Chang et al 1993(Chang et al , 1996, while they are polynomial-time solvable for balanced graphs (Bonomo et al 2006a;Dahlhaus et al 1998), comparability graphs (Balachandran et al 1996), distance-hereditary graphs (Lee and Chang 2006), Helly circular-arc graphs (Guruswami and Rangan 2000;Durán et al 2008), doubly chordal graphs (Brandstädt et al 1997), and strongly chordal graphs (Brandstädt et al 1997;Chang et al 1993Chang et al , 1996. For the clique transversal set problem, two examples of applications in communications and social networks were presented in Xu et al (2007). Clique perfect graphs have been studied in many articles, too (Balachandran et al 1996;Berge and Las Vergnas 1970;Bonomo et al 2006aBonomo et al , 2006bBonomo et al , 2008Bonomo et al , 2009Chang et al 1993;Lee and Chang 2006;Lehel and Tuza 1986).…”

Section: For Any Vertexmentioning

confidence: 98%

“…A maximum-clique transversal set can be viewed as a Y -maximum-clique transversal function with Y = {0, 1}. In this case, the function f is called a maximum-clique transversal function of G. Then, the maximum-clique transversal number of G is equal to the minimum weight of a maximumclique transversal function of G. The concept was also used in Xu et al (2007). It motivates us to study the following variations of the maximum-clique transversal set problem.…”

Section: For Any Vertexmentioning

confidence: 98%

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Variations of maximum-clique transversal sets on graphs

Lee

2009

Ann Oper Res

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A maximum-clique transversal set of a graph G is a subset of vertices intersecting all maximum cliques of G. The maximum-clique transversal set problem is to find a maximum-clique transversal set of G of minimum cardinality. Motivated by the placement of transmitters for cellular telephones, Chang, Kloks, and Lee introduced the concept of maximum-clique transversal sets on graphs in 2001. In this paper, we introduce the concept of maximum-clique perfect and some variations of the maximum-clique transversal set problem such as the {k}-maximum-clique, k-fold maximum-clique, signed maximumclique, and minus maximum-clique transversal problems. We show that balanced graphs, strongly chordal graphs, and distance-hereditary graphs are maximum-clique perfect. Besides, we present a unified approach to these four problems on strongly chordal graphs and give complexity results for the following classes of graphs:

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Section: For Any Vertexmentioning

confidence: 98%

Section: For Any Vertexmentioning

confidence: 98%

Section: For Any Vertexmentioning

confidence: 98%

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Variations of maximum-clique transversal sets on graphs

Lee

2009

Ann Oper Res

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“…Remark 4 Xu, et al [56] studied minus cliquetransversal that can be regarded as a special case of minus domination.…”

Section: Minus Domination In Graphsmentioning

confidence: 99%

Dominating functions with integer values in graphs—a survey

Kang

Shan

2007

J. of Shanghai Univ.

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For an arbitrary subset P of the reals, a function f : V → P is defined to be a P-dominating function of a graph G = (V, E) if the sum of its function values over any closed neighbourhood is at least 1. That is, for everyThe definition of total P-dominating function is obtained by simply changing 'closed' neighborhood N [v] in the definition of P-dominating function to 'open' neighborhood N (v). The (total) P-domination number of a graph G is defined to be the infimum of weight w(f ) = P v∈V f (v) taken over all (total) P-dominating function f . Similarly, the P-edge and P-star dominating functions can be defined. In this paper we survey some recent progress on the topic of dominating functions in graph theory. Especially, we are interested in P-, P-edge and P-star dominating functions of graphs with integer values.

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“…So far, no more progress has been made on the above problem. For else investigations on the clique-transversal number of graphs, we refer to [14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Bounds on the clique-transversal number of regular graphs

Shan

Cheng

Kang

2008

Sci. China Ser. A-Math.

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A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted τc(G), is the minimum cardinality of a cliquetransversal set in G. In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound. Also, we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.

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The algorithmic complexity of the minus clique-transversal problem

Cited by 11 publications

References 14 publications

Variations of maximum-clique transversal sets on graphs

Variations of maximum-clique transversal sets on graphs

Dominating functions with integer values in graphs—a survey

Bounds on the clique-transversal number of regular graphs

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