Clique-Width of Graph Classes Defined by Two Forbidden Induced Subgraphs

Dabrowski, Konrad K.; Paulusma, Daniël

doi:10.1093/comjnl/bxv096

Cited by 39 publications

(71 citation statements)

References 58 publications

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“…However, already when one considers H ‐free weakly chordal graphs, one does not obtain any new tractable graph classes. Indeed, the clique‐width of the class of H ‐free graphs is bounded if and only if H is an induced subgraph of P 4 , and as we prove later, the induced subgraphs of P 4 are also the only graphs H for which the class of H ‐free weakly chordal graphs has bounded clique‐width. The same classification therefore also follows for superclasses, such as

(H, C_{5}, C_{6}, \dots)

‐free graphs (or H ‐free perfect graphs, to give another example).…”

Section: Introductionsupporting

confidence: 52%

“…Including our new result for the case

(2 P_{1} + P_{3}, K_{4})

and five cases recently proved by Dabrowski et al. , this led to a classification of all but eight open cases (up to some equivalence relation, see ). Complete a line of research on H ‐ free chordal graphs .A classification of those graphs H for which the clique‐width of H ‐free chordal graphs is bounded would complete a line of research in the literature, which we feel is an interesting goal on its own. As a start, using a result of Corneil and Rotics on the relationship between treewidth and clique‐width, it follows that the clique‐width of the class of

K_{r}

‐free chordal graphs is bounded for all

r \geq 1

.…”

Section: Introductionmentioning

confidence: 57%

“…In particular, one of these three classes, namely the class of

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

‐free graphs was reduced to the class of

\bar{2 P_{1} + P_{2}}

‐free chordal graphs, also known as diamond‐free chordal graphs (the diamond is the graph

\bar{2 P_{1} + P_{2}}

, see also Fig. ), which has bounded clique‐width .Our new result for the class of

(2 P_{1} + P_{3}, K_{4})

‐free graphs and the three results of belong to a line of research, trying to extend results on the clique‐width of classes of

(H_{1}, H_{2})

‐free graphs in order to try to determine the boundedness or unboundedness of the clique‐width of every such graph class . Including our new result for the case

(2 P_{1} + P_{3}, K_{4})

and five cases recently proved by Dabrowski et al.…”

Section: Introductionmentioning

confidence: 60%

“…Classifying the boundedness of clique‐width for H ‐free chordal graphs turns out to be useful for determining the (un)boundedness of the clique‐width of graph classes characterized by two forbidden induced subgraphs H 1 and H 2 , just as the full classification for H ‐free bipartite graphs has proven to be . To demonstrate this, we will prove that the class of

(2 P_{1} + P_{3}, K_{4})

‐free graphs has bounded clique‐width via a reduction to K 4 ‐free chordal graphs.…”

Section: Introductionmentioning

confidence: 97%

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Bounding the Clique‐Width of H‐Free Chordal Graphs

Brandstädt

Dabrowski

Huang

et al. 2017

Journal of Graph Theory

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Abstract:A graph is H-free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le, and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le, and Mosca erroneously claimed that the gem and co-gem are the only two 1-vertex P 4 -extensions H for which the class of H-free chordal graphs has bounded clique-width. In fact we prove that bull-free chordal and co-chair-free chordal graphs have clique-width at most 3 and 4, respectively. In particular, we find four new classes of H-free chordal graphs 42This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. BOUNDING CLIQUE-WIDTH OF H-FREE CHORDAL GRAPHS 43of bounded clique-width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H-free chordal graphs has bounded clique-width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of (2P 1 + P 3 , K 4 )-free graphs has bounded clique-width via a reduction to K 4 -free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique-width of H-free weakly chordal graphs.

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(H, C_{5}, C_{6}, \dots)

‐free graphs (or H ‐free perfect graphs, to give another example).…”

Section: Introductionsupporting

confidence: 52%

“…Including our new result for the case

(2 P_{1} + P_{3}, K_{4})

K_{r}

‐free chordal graphs is bounded for all

r \geq 1

.…”

Section: Introductionmentioning

confidence: 57%

“…In particular, one of these three classes, namely the class of

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

‐free graphs was reduced to the class of

\bar{2 P_{1} + P_{2}}

‐free chordal graphs, also known as diamond‐free chordal graphs (the diamond is the graph

\bar{2 P_{1} + P_{2}}

, see also Fig. ), which has bounded clique‐width .Our new result for the class of

(2 P_{1} + P_{3}, K_{4})

‐free graphs and the three results of belong to a line of research, trying to extend results on the clique‐width of classes of

(H_{1}, H_{2})

‐free graphs in order to try to determine the boundedness or unboundedness of the clique‐width of every such graph class . Including our new result for the case

(2 P_{1} + P_{3}, K_{4})

and five cases recently proved by Dabrowski et al.…”

Section: Introductionmentioning

confidence: 60%

(2 P_{1} + P_{3}, K_{4})

‐free graphs has bounded clique‐width via a reduction to K 4 ‐free chordal graphs.…”

Section: Introductionmentioning

confidence: 97%

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Bounding the Clique‐Width of H‐Free Chordal Graphs

Brandstädt

Dabrowski

Huang

et al. 2017

Journal of Graph Theory

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“…Then they applied Theorem . Dabrowski and Paulusma proved that the class of

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

‐free graphs has bounded clique width, so Theorem (i) can be applied. 4.Theorem implies that for all

r \geq 1

, Coloring is polynomial‐time solvable on

(K_{r}, F)

‐free graphs for some linear forest F if k ‐ Coloring is polynomial‐time solvable on F ‐free graphs for all

k \geq 1

.…”

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

confidence: 99%

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

Golovach

Johnson

Paulusma

et al. 2016

Journal of Graph Theory

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130

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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