Path parity and perfection

Everett, Hazel; Figueiredo, Celina M. H. de; Sales, Cláudia Linhares; Maffray, Frédéric; Porto, Oscar; Reed, Bruce

doi:10.1016/s0012-365x(96)00174-4

Cited by 26 publications

(24 citation statements)

References 28 publications

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“…Strongly perfect graphs were introduced by Berge and Duchet in 1984 [4]. It is known that the following graphs are strongly perfect: -perfectly orderable graphs [5], [6]; -graphs with Dilworth number at most 3 [12]; -(quasi-) Meyniel graphs [10], [7].…”

Section: Former Results On Sufficient Conditions Formentioning

confidence: 99%

Coloring the cliques of line graphs

Bacsó

Ryjáček

Tuza

2017

Discrete Mathematics

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The weak chromatic number , or clique chromatic number (CCHN) of a graph is the minimum number of colors in a vertex coloring, such that every maximal clique gets at least two colors. The weak chromatic index , or clique chromatic index (CCHI) of a graph is the CCHN of its line graph.Most of the results here are upper bounds for the CCHI, as functions of some other graph parameters, and contrasting with lower bounds in some cases. Algorithmic aspects are also discussed; the main result within this scope (and in the paper) shows that testing whether the CCHI of a graph equals 2 is NP-complete. We deal with the CCHN of the graph itself as well.

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Section: Former Results On Sufficient Conditions Formentioning

confidence: 99%

Coloring the cliques of line graphs

Bacsó

Ryjáček

Tuza

2017

Discrete Mathematics

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“…If a graph G has an even pair, then by contracting it into a single vertex, we do not change the clique number of G. Moreover, under this transformation the chromatic number of G does not change as well (see e.g. [15]). …”

Section: Transformations For Related Problemsmentioning

confidence: 99%

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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“…Later Meyniel showed that minimal non-perfect graphs contain no even pair [29]. Those two facts triggered a series of theoretical and algorithmic results which are surveyed in [16] and its updated version [17].…”

Section: Odd Induced Pathmentioning

confidence: 99%

Finding Induced Paths of Given Parity in Claw-Free Graphs

2010

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Citation for published item:rof vn 9tD F nd umi¡ nskiD wF nd ulusmD hniel @PHIPA 9pinding indued pths of given prity in lwEfree grphsF9D elgorithmiFD TP @IEPAF ppF SQUESTQF Further information on publisher's website:The nal publication is available at Springer via http://dx.doi.org/10.1007/s00453-010-9470-5Additional 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. The Parity Path problem is to decide if a given graph contains both an induced path of odd length and an induced path of even length between two specified vertices. In the related problems Odd Induced Path and Even Induced Path, the goal is to determine whether an induced path of odd, respectively even, length between two specified vertices exists. Although all three problems are NP-complete in general, we show that they can be solved in O(n 5 ) time for the class of claw-free graphs. Two vertices s and t form an even pair in G if every induced path from s to t in G has even length. Our results imply that the problem of deciding if two specified vertices of a claw-free graph form an even pair, as well as the problem of deciding if a given claw-free graph has an even pair, can be solved in O(n 5 ) time and O(n 7 ) time, respectively. We also show that we can decide in O(n 7 ) time whether a claw-free graph has an induced cycle of given parity through a specified vertex. Finally, we show that a shortest induced path of given parity between two specified vertices of a claw-free perfect graph can be found in O(n 7 ) time.

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Path parity and perfection

Cited by 26 publications

References 28 publications

Coloring the cliques of line graphs

Coloring the cliques of line graphs

From matchings to independent sets

Finding Induced Paths of Given Parity in Claw-Free Graphs

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