Two cases of polynomial-time solvability for the coloring problem

Malyshev, D. S.

doi:10.1007/s10878-014-9792-3

Cited by 29 publications

(14 citation statements)

References 13 publications

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“…">11.The class of

(4 P_{1}, \bar{2 P_{1} + P_{3}})

‐free graphs has bounded clique width , hence we apply Theorem (i). 12.This was proved by Malyshev for

(P_{5}, C_{4})

and by Malyshev for

(P_{5}, \bar{2 P_{1} + P_{3}})

□

…”

Section: Results and Open Problems For (H1h2)‐free Graphsmentioning

confidence: 65%

“…">1.This follows from Theorem (i). 2.This was proved by Malyshev for

(K_{1, 3}, C_{3}^{*})

‐free graphs and

(K_{1, 3}, P_{5})

‐free graphs and by Malyshev for

(K_{1, 3}, C_{3}^{+ +})

‐free graphs. 3.First we consider the case when H1 is a forest on at most six vertices not isomorphic to K 1, 5 and

H_{2} \subseteq_{i} C_{3}

.…”

Section: Results and Open Problems For (H1h2)‐free Graphsmentioning

confidence: 90%

“…2. This was proved by Malyshev [84] for (K 1,3 , C * 3 )-free graphs and (K 1,3 , P 5 )free graphs and by Malyshev [86] for (K 1,3 , C ++ 3 )-free graphs. 3.…”

Section: Theorem 21 Let H 1 and H 2 Be Two Graphs Then The Followinmentioning

confidence: 79%

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A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

Golovach

Johnson

Paulusma

et al. 2016

Journal of Graph Theory

132

160

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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. For a positive integer k, a k-colouring of a graph G = (V, E) is a mapping c : V → {1, 2, . . . , k} such that c(u) = c(v) whenever uv ∈ E. The COLOURING problem is to decide, for a given G and k, whether a k-colouring of G exists. If k is fixed (that is, it is not part of the input), we have the decision problem k-COLOURING instead. We survey known results on the computational complexity of COLOURING and k-COLOURING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex. Finally, we also survey results for graph classes defined by some other forbidden pattern.

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“…">11.The class of

(4 P_{1}, \bar{2 P_{1} + P_{3}})

‐free graphs has bounded clique width , hence we apply Theorem (i). 12.This was proved by Malyshev for

(P_{5}, C_{4})

and by Malyshev for

(P_{5}, \bar{2 P_{1} + P_{3}})

□

…”

Section: Results and Open Problems For (H1h2)‐free Graphsmentioning

confidence: 65%

“…">1.This follows from Theorem (i). 2.This was proved by Malyshev for

(K_{1, 3}, C_{3}^{*})

‐free graphs and

(K_{1, 3}, P_{5})

‐free graphs and by Malyshev for

(K_{1, 3}, C_{3}^{+ +})

‐free graphs. 3.First we consider the case when H1 is a forest on at most six vertices not isomorphic to K 1, 5 and

H_{2} \subseteq_{i} C_{3}

.…”

Section: Results and Open Problems For (H1h2)‐free Graphsmentioning

confidence: 90%

See 1 more Smart Citation

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

Golovach

Johnson

Paulusma

et al. 2016

Journal of Graph Theory

132

160

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“…Both polynomial‐time and

sans-serif𝖭𝖯

‐completeness results are known for

(H_{1}, H_{2})

‐free graphs, as shown by many teams of researchers (see e.g. ), but the complexity classification is far from complete: even if we forbid two graphs

H_{1}

and

H_{2}

of up to four vertices, there are still three open cases left, namely when

(H_{1}, H_{2}) \in {(K_{1, 3}, 4 P_{1}), (K_{1, 3}, 2 P_{1} + P_{2}), (C_{4}, 4 P_{1})}

; see (the graph

K_{1, 3}

is the claw and

C_{4}

is the

4

‐vertex cycle). We refer to [18, Theorem 21] for a summary and to the recent paper , in which the number of missing cases when

H_{1}

and

H_{2}

are both connected graphs on at most 5 vertices was reduced from 10 to 8.…”

Section: Introductionmentioning

confidence: 99%

Hereditary graph classes: When the complexities of coloring and clique cover coincide

Blanché

Dabrowski

Johnson

et al. 2018

Journal of Graph Theory

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A graph is ( H 1 , H 2 )‐free for a pair of graphs H 1 , H 2 if it contains no induced subgraph isomorphic to H 1 or H 2. In 2001, Král et al initiated a study into the complexity of coloring for ( H 1 , H 2 )‐free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those ( H 1 , H 2 )‐free graphs where H 2 is H true¯ 1, the complement of H 1. As these classes are closed under complementation, the computational complexities of coloring and clique cover coincide. By combining new and known results, we are able to classify the complexity of coloring and clique cover for ( H , trueH ¯ )‐free graphs for all cases except when H = s P 1 + P 3 for s ≥ 3 or H = s P 1 + P 4 for s ≥ 2. We also classify the complexity of coloring on graph classes characterized by forbidding a finite number of self‐complementary induced subgraphs, and we initiate a study of k‐ coloring for ( P r , P true¯ r )‐free graphs.

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“…Recently, the complexity dichotomy of the VC problem for classes of graphs which are defined by two connected five-vertex forbidden induced subgraphs received considerable attention. The VC problem is known to be solvable in polynomial time for: (P 5 , gem)-free graphs [1], (P 5 , P 5 )-free graphs [18], (P 5 , P 3 + O 2 )-free graphs [28], (P 5 , P 3 + P 2 )-free graphs [29], and for (P 5 , K 5 − e)-free graphs [29]. In particular, the complexity dichotomy of the VC problem is known for classes of graphs which are defined by two connected five-vertex forbidden induced subgraphs except for the following seven cases: (fork, bull)-free graphs, and (P 5 , H)-free graphs, where H ∈ {K 3 + O 2 , K 2,3 , dart, banner, bull, 2P 2 + P 1 }.…”

Section: Introductionmentioning

confidence: 99%

Polynomial Cases for the Vertex Coloring Problem

Karthick¹,

Maffray²,

Pastor³

2018

Algorithmica

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Two cases of polynomial-time solvability for the coloring problem

Abstract: The complexity of the coloring problem is known for all hereditary classes defined by two connected 5-vertex forbidden induced subgraphs except 13 cases. We update this result by proving polynomial-time solvability of the problem for two of the mentioned 13 classes.

Cited by 29 publications

References 13 publications

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

Hereditary graph classes: When the complexities of coloring and clique cover coincide

Polynomial Cases for the Vertex Coloring Problem

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