Algorithms for finding clique-transversals of graphs

Durán, Guillermo; Lin, Min Chih; Mera, Sergio; Szwarcfiter, Jayme Luiz

doi:10.1007/s10479-007-0189-x

Cited by 13 publications

(9 citation statements)

References 15 publications

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“…See the recent work of Shan et al [52] and references therein. In computer science, exactly computing the smallest set on special classes of graphs appears in many works [35,46,13,27,47].…”

Section: Cliquesmentioning

confidence: 99%

Inapproximability of $H$-Transversal/Packing

Guruswami

Lee

2017

SIAM J. Discrete Math.

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Given an undirected graph G = (V G , E G ) and a fixed "pattern" graph H = (V H , E H ) with k vertices, we consider the H-Transversal and H-Packing problems. The former asks to find the smallest S ⊆ V G such that the subgraph induced by V G \ S does not have H as a subgraph, and the latter asks to find the maximum number of pairwise disjoint k-subsets S 1 , ..., S m ⊆ V G such that the subgraph induced by each S i has H as a subgraph.We prove that if H is 2-connected, H-Transversal and H-Packing are almost as hard to approximate as general k-Hypergraph Vertex Cover and k-Set Packing, so it is NP-hard to approximate them within a factor of Ω(k) and Ω(k) respectively. We also show that there is a 1-connected H where H-Transversal admits an O(log k)-approximation algorithm, so that the connectivity requirement cannot be relaxed from 2 to 1. For a special case of H-Transversal where H is a (family of) cycles, we mention the implication of our result to the related Feedback Vertex Set problem, and give a different hardness proof for directed graphs. * It is an expanded, generalized, and refocused version of our earlier unpublished manuscript [33] and our conference version that appears in the proceedings of APPROX 15. †

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“…See the recent work of Shan et al [52] and references therein. In computer science, exactly computing the smallest set on special classes of graphs appears in many works [35,46,13,27,47].…”

Section: Cliquesmentioning

confidence: 99%

Inapproximability of $H$-Transversal/Packing

Guruswami

Lee

2017

SIAM J. Discrete Math.

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“…Trotter [70] has proved that a partial order of a CA comparability graph has dimension at most 4. Durán, Lin, Mera and Szwarcfiter [12,13] have described polynomial time algorithms for finding maximum clique-independent sets and minimum clique-transversals in 3K 2 -free CA graphs. Lin, McConnell, Soulignac and Szwarcfiter [39] have studied the clique structure of HCA graphs.…”

Section: Other Classesmentioning

confidence: 99%

Characterizations and recognition of circular-arc graphs and subclasses: A survey

Lin

Szwarcfiter

2009

Discrete Mathematics

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a b s t r a c tCircular graphs are intersection graphs of arcs on a circle. These graphs are reported to have been studied since 1964, and they have been receiving considerable attention since a series of papers by Tucker in the 1970s. Various subclasses of circular-arc graphs have also been studied. Among these are the proper circular-arc graphs, unit circular-arc graphs, Helly circular-arc graphs and co-bipartite circular-arc graphs. Several characterizations and recognition algorithms have been formulated for circular-arc graphs and its subclasses. In particular, it should be mentioned that linear time algorithms are known for all these classes of graphs. In the present paper, we survey these characterizations and recognition algorithms, with emphasis on the linear time algorithms.

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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: 99%

“…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).…”

Section: For Any Vertexmentioning

confidence: 99%

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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Algorithms for finding clique-transversals of graphs

Cited by 13 publications

References 15 publications

Inapproximability of $H$-Transversal/Packing

Inapproximability of $H$-Transversal/Packing

Characterizations and recognition of circular-arc graphs and subclasses: A survey

Variations of maximum-clique transversal sets on graphs

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