On easy and hard hereditary classes of graphs with respect to the independent set problem

Alekseev, V. E.

doi:10.1016/s0166-218x(03)00387-1

Cited by 87 publications

(76 citation statements)

References 10 publications

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“…This theorem was proved by Alekseev in [3]. Moreover, in the same paper Alekseev proved that the class S is a minimal limit class, i.e.…”

Section: Let Us Denote Bymentioning

confidence: 80%

“…In the next section, we study the maximum independent set problem. 3 The maximum independent set problem on hereditary classes of graphs…”

Section: Hereditary Classes Of Graphsmentioning

confidence: 99%

“…In other words, limit classes play the role of "forbidden elements" for the family of finitely defined classes with polynomial-time solvable independent set problem. Similarly to the induced subgraph characterization of hereditary classes, only minimal limit classes are of interest, which justifies the following definition introduced in [3]. Definition 2.…”

Section: Let Us Denote Bymentioning

confidence: 99%

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From matchings to independent sets

Lozin

2017

Discrete Applied Mathematics

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In 1965, Jack Edmonds proposed his celebrated polynomial-time algorithm to find a maximum matching in a graph. It is well-known that finding a maximum matching in G is equivalent to finding a maximum independent set in the line graph of G. For general graphs, the maximum independent set problem is NP-hard. What makes this problem easy in the class of line graphs and what other restrictions can lead to an efficient solution of the problem? In the present paper, we analyze these and related questions. We also review various techniques that may lead to efficient algorithms for the maximum independent set problem in restricted graph families, with a focus given to augmenting graphs and graph transformations. Both techniques have been used in the solution of Edmonds to the maximum matching problem, i.e. in the solution to the maximum independent set problem in the class of line graphs. We survey various results that exploit these techniques beyond the line graphs.

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“…This theorem was proved by Alekseev in [3]. Moreover, in the same paper Alekseev proved that the class S is a minimal limit class, i.e.…”

Section: Let Us Denote Bymentioning

confidence: 80%

“…In the next section, we study the maximum independent set problem. 3 The maximum independent set problem on hereditary classes of graphs…”

Section: Hereditary Classes Of Graphsmentioning

confidence: 99%

Section: Let Us Denote Bymentioning

confidence: 99%

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From matchings to independent sets

Lozin

2017

Discrete Applied Mathematics

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“…In contrast, already 1-Contraction Blocker(α) is NP-complete for P 5 -free graphs (recall that it is NP-complete even for cobipartite graphs, as explained in Section 1). The problems of determining the chromatic number [19] and the clique number [1] are NP-hard for P 5 -free graphs. One might be able to use these two results to prove NP-hardness of d-Contraction Blocker(π) for π ∈ {χ, ω} and d ≥ 1.…”

Section: Discussionmentioning

confidence: 99%

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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“…Surprisingly we show that these problems and several of their variants that are known to be hard in the W-hierarchy, are fixed parameter tractable on graphs that have no short cycles -more specifically on graphs with girth at least five. These problems are known to be NP-complete on such graphs as well [4,5]. We also look at the Set Cover problem where the size of the intersection of any pair of sets is bounded by a fixed constant.…”

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confidence: 99%

Short Cycles Make W-hard Problems Hard: FPT Algorithms for W-hard Problems in Graphs with no Short Cycles

2008

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Abstract. We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles.More specifically, we give fixed parameter tractable algorithms for Dominating Set, t-Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W [i]-hard for some i ≥ 1 in general graphs. We also show that the Dominating Set problem is W [2]-hard for bipartite graphs and hence for triangle free graphs.In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, paramaterized by k, is W [1]-hard even on graphs with girth at least six.Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph.

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On easy and hard hereditary classes of graphs with respect to the independent set problem

Cited by 87 publications

References 10 publications

From matchings to independent sets

From matchings to independent sets

Contraction Blockers for Graphs with Forbidden Induced Paths

Short Cycles Make W-hard Problems Hard: FPT Algorithms for W-hard Problems in Graphs with no Short Cycles

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