The strong perfect graph theorem

Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin

doi:10.4007/annals.2006.164.51

Cited by 1,035 publications

(1,033 citation statements)

References 19 publications

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“…Hence, G * contains no induced hole of odd size and no induced antihole of odd size, where a hole is an induced cycle on at least five vertices and an antihole is the complement of a hole. Then, by the Strong Perfect Graph Theorem [5], G * is perfect as well.…”

Section: Lemma 7 ([7]mentioning

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: Lemma 7 ([7]mentioning

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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“…A hole is even (odd) if it has even (odd) length. In the last time problems concerning holes have received much attention as they are related to the Strong Perfect Graph Theorem ("A graph is perfect if it contains neither an odd hole nor the complement of an odd hole"), which has been proven recently [6,8]. We mention some results and open problems: It is not known whether there is a polynomial time algorithm deciding if a graph has an odd hole, while the questions whether a graph contains a hole and whether it contains an even hole are solvable in polynomial time (cf.…”

Section: Holesmentioning

confidence: 99%

“…Recently, the search for algorithms detecting chordless cycles (of odd length ≥ 5) has received much attention due to its relationship to Berge graphs and to the Strong Perfect Graph Theorem (cf. [6,8,10,11,26]). …”

Section: Introductionmentioning

confidence: 99%

On Parameterized Path and Chordless Path Problems

Chen

Flum

2007

Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07)

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We study the parameterized complexity of various path (and cycle) problems, the parameter being the length of the path. For example, we show that the problem of the existence of a maximal path of length k in a graph G is fixed-parameter tractable, while its counting version is #W[1]-complete. The corresponding problems for chordless (or induced) paths are W[2]-complete and #W[2]-complete respectively. With the tools developed in this paper we derive the NP-completeness of a related classical problem, thereby solving a problem due to Hedetniemi [21].

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“…Another characterization of a perfect graph is via forbidden subgraphs: a graph is perfect if and only if it does not have odd holes (induced cycles of odd length at least 5) or odd anti-holes (the complement graph of an odd hole) [10]. Both odd holes and odd anti-holes are connected graphs.…”

Section: Example -Perfect Graphsmentioning

confidence: 99%

Fast Distributed Algorithms for Testing Graph Properties

Censor-Hillel

Fischer

Schwartzman

et al. 2016

Lecture Notes in Computer Science

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We initiate a thorough study of distributed property testing -producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called dense graph testing model we emulate sequential tests for nearly all graph properties having 1-sided tests, while in the general and sparse models we obtain faster tests for trianglefreeness, cycle-freeness and bipartiteness, respectively. In addition, we show a logarithmic lower bound for testing bipartiteness and cycle-freeness, which holds even in the stronger LOCAL model.In most cases, aided by parallelism, the distributed algorithms have a much shorter running time as compared to their counterparts from the sequential querying model of traditional property testing. The simplest property testing algorithms allow a relatively smooth transitioning to the distributed model. For the more complex tasks we develop new machinery that may be of independent interest.

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The strong perfect graph theorem

Cited by 1,035 publications

References 19 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

On Parameterized Path and Chordless Path Problems

Fast Distributed Algorithms for Testing Graph Properties

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