The most vital nodes with respect to independent set and vertex cover

Bazgan, Cristina; Toubaline, Sonia; Tuza, Źsolt

doi:10.1016/j.dam.2011.06.023

Cited by 48 publications

(56 citation statements)

References 15 publications

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“…In the same section we complement this result by showing that Contraction Blocker(α) can be solved in linear time for trees. Then, in Section 4, we prove that Deletion Blocker(α) is co-NP-hard for triangle-free graphs even if d = k = 1 (in contrast the problem is polynomial-time solvable for bipartite graphs [2,5]). In Section 5 we extend our result for triangle-free graphs to other graph classes for which Independent Set is NP-complete.…”

Section: Our Resultsmentioning

confidence: 99%

Blocking Independent Sets for H-Free Graphs via Edge Contractions and Vertex Deletions

Paulusma

Picouleau

Ries

2017

Lecture Notes in Computer Science

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Citation for published item:ulusmD hni¤ el nd iouleuD ghristophe nd iesD fernrd @PHIUA 9floking independent sets for rEfree grphs vi edge ontrtions nd vertex deletionsF9D in heory nd epplitions of wodels of gomputtionD IRth ennul gonferene @ewg PHIUA X fernD witzerlndD epril PHEPPD PHIU Y proeedingsF ghmX pringerD ppF RUHERVQF veture xotes in gomputer ieneF @IHIVSAF Further information on publisher's website:Publisher's copyright statement:The nal publication is available at Springer via https://doi.org/10.1007/978-3-319-55911-734Additional information: Use policyThe 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-prot 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. Let d and k be two given integers, and let G be a graph.Can we reduce the independence number of G by at least d via at most k graph operations from some fixed set S? This problem belongs to a class of so-called blocker problems. It is known to be co-NP-hard even if S consists of either an edge contraction or a vertex deletion. We further investigate its computational complexity under these two settings: -we give a sufficient condition on a graph class for the vertex deletion variant to be co-NP-hard even if d = k = 1; -in addition we prove that the vertex deletion variant is co-NP-hard for triangle-free graphs even if d = k = 1; -we prove that the edge contraction variant is NP-hard for bipartite graphs but linear-time solvable for trees. By combining our new results with known ones we are able to give full complexity classifications for both variants restricted to H-free graphs.

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Section: Our Resultsmentioning

confidence: 99%

Blocking Independent Sets for H-Free Graphs via Edge Contractions and Vertex Deletions

Paulusma

Picouleau

Ries

2017

Lecture Notes in Computer Science

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“…For edge additions the problem, for π = α, is known [2] to be NP-hard for general graphs even if d = 1 and polynomial-time solvable on split graphs if d is fixed. For vertex deletions the problem, for π = ω, is known to be NP-complete for general graphs [21] and, for π = α, polynomialtime solvable for cographs if d is part of the input [4]. It would be interesting to complete these results in the way we have done for edge contractions.…”

Section: Theorem 5 (♠)mentioning

confidence: 95%

Contraction Blockers for Graphs with Forbidden Induced Paths

Diner

Paulusma

Picouleau

et al. 2015

Lecture Notes in Computer Science

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hinerD ¤ yF nd ulusm D hF nd i oule uD gF nd iesD fF @PHISA 9gontr tion lo kers for gr phs with for idden indu ed p thsF9D in elgorithms nd omplexity X Wth sntern tion l gonferen eD gseg PHISD risD pr n eD w y PHEPPD PHIS Y pro eedingsF D ppF IWREPHUF ve ture notes in omputer s ien eF @WHUWAF Further information on publisher's website:Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-18173-814.Additional information: Use policyThe 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 be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pfree graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomialtime solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs.

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“…The purpose is to find a subset of vertices whose removal from G results in the greatest decrease in the weight of the maximum independent set. Bazgan et al [1] show that this problem is NP-hard on the bipartite graphs and is polynomial on the unweighted bipartite graphs.…”

Section: Introductionmentioning

confidence: 99%

“…The interdiction problem has discussed on some basic problems such as the maximum flow network, shortest path and so on [2,4]. This problem has received some attention in recent years and has expressed on the problems of graph theory such as the vertex cover, independent set, connectivity, matching and so on [1,5,6]. Shen et al [5] introduced an interdiction problem that disconnectivity of a graph is maximized by deleting a subset of vertices.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Independent Set Interdiction Problem

Shirdel¹,

Kahkeshani²

2015

ejgta

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The purpose of the independent set interdiction problem in the weighted graph G is to determine a set of vertices R * such that the weight of the maximum independent set in G − R * is minimized. We define an approximate solution for this problem. Then, an upper bound for the relative error of this problem is obtained. We show that the limit of the relative error of the independent set interdiction problem in some subclasses of the generalized Petersen graphs is zero as the number of vertices tends to infinity.

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The most vital nodes with respect to independent set and vertex cover

Cited by 48 publications

References 15 publications

Blocking Independent Sets for H-Free Graphs via Edge Contractions and Vertex Deletions

Blocking Independent Sets for H-Free Graphs via Edge Contractions and Vertex Deletions

Contraction Blockers for Graphs with Forbidden Induced Paths

On the Independent Set Interdiction Problem

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