Blockers and transversals in some subclasses of bipartite graphs: When caterpillars are dancing on a grid

Ries, Bernard; Bentz, Cédric; Picouleau, Christophe; Werra, Dominique de; Costa, Marie-Christine; Zenklusen, Rico

doi:10.1016/j.disc.2009.08.009

Cited by 37 publications

(22 citation statements)

References 6 publications

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“…Identifying the part of the graph responsible for a significant decrease of the graph parameter under consideration gives crucial information on the graph. Blocker problems have been given much attention over the last years [1- 4,6,7,16,18,19]. Graph parameters considered were the chromatic number, the independence number, the clique number, the matching number and the vertex cover number.…”

Section: Introductionmentioning

confidence: 99%

Reducing the Clique and Chromatic Number via Edge Contractions and Vertex Deletions

Paulusma

Picouleau

Ries

2016

Lecture Notes in Computer Science

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. We consider the following problem: can a certain graph parameter of some given graph G be reduced by at least d, for some integer d, via at most k graph operations from some specified set S, for some given integer k? As graph parameters we take the chromatic number and the clique number. We let the set S consist of either an edge contraction or a vertex deletion. As all these problems are NP-complete for general graphs even if d is fixed, we restrict the input graph G to some special graph class. We continue a line of research that considers these problems for subclasses of perfect graphs, but our main results are full classifications, from a computational complexity point of view, for graph classes characterized by forbidding a single induced connected subgraph H.

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Section: Introductionmentioning

confidence: 99%

Reducing the Clique and Chromatic Number via Edge Contractions and Vertex Deletions

Paulusma

Picouleau

Ries

2016

Lecture Notes in Computer Science

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“…In this paper, we introduce the problem of detecting a smallest subset of edges in a weighted undirected graph whose removal bounds the minimum weight of a dominating set in the remaining graph. Adopting the terminology used in the literature for similar problems (Bazgan et al, 2011;Ries et al, 2010;Zenklusen et al, 2009), this problem will be referred to as the minimum edge blocker dominating set problem (EBDP). To avoid excessively long acronyms, we use EBDP rather than MEBDP, since the only non-trivial setup of the considered problem is minimization.…”

Section: Introductionmentioning

confidence: 99%

“…In the related literature, the first problem is referred to as minimum vertex/edge blocker problem (Bazgan, Toubaline, & Tuza, 2011;Ries, Bentz, Picouleau, de Werra, Costa, & Zenklusen, 2010;Zenklusen, Ries, Picouleau, de Werra, Costa, & Bentz, 2009) and the second one is called most vital vertices/edges problem (Bar-Noy, Khuller, & Schieber, 1995;Bazgan et al, 2011;Bazgan, Toubaline, & Vanderpooten, 2010a). In this paper, we introduce the problem of detecting a smallest subset of edges in a weighted undirected graph whose removal bounds the minimum weight of a dominating set in the remaining graph.…”

Section: Introductionmentioning

confidence: 99%

Minimum edge blocker dominating set problem

Pajouh

Walteros

Boginski

et al. 2015

European Journal of Operational Research

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“…It has been shown that finding a minimum d-transversal of all maximum matchings in a bipartite graph is N P -hard for any fixed d ≥ 1 [25], while it can be found in polynomial time in trees and grid graphs [18]. In [16], Lee reviews and gives some new results for the minimum d-transversal problem in the case where C V is the set of maximum cliques of a given graph G; a d-transversal is called there a d-fold maximum clique transversal set.…”

Section: Preliminariesmentioning

confidence: 99%

Minimum d-blockers and d-transversals in graphs

2010

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We consider a set V of elements and an optimization problem on V : the search for a maximum (or minimum) cardinality subset of V verifying a given property P. A d-transversal is a subset of V which intersects any optimum solution in at least d elements while a d-blocker is a subset of V whose removal deteriorates the value of an optimum solution by at least d. We present some general characteristics of these problems, we review some situations which have been studied (matchings, s − t paths and s − t cuts in graphs) and we study d-transversals and d-blockers for new problems as stable sets or vertex covers in bipartite graphs.

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Blockers and transversals in some subclasses of bipartite graphs: When caterpillars are dancing on a grid

Cited by 37 publications

References 6 publications

Reducing the Clique and Chromatic Number via Edge Contractions and Vertex Deletions

Reducing the Clique and Chromatic Number via Edge Contractions and Vertex Deletions

Minimum edge blocker dominating set problem

Minimum d-blockers and d-transversals in graphs

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