Bounding clique-width via perfect graphs

Dabrowski, Konrad K.; Huang, Shenwei; Paulusma, Daniël

doi:10.1016/j.jcss.2016.06.007

Cited by 12 publications

(15 citation statements)

References 31 publications

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“…Towards our first goal we prove in Section 4 that the class of (2P 1 + P 2 , P 2 + P 3 )-free graphs (which has bounded clique-width [11]) is well-quasi-ordered by the induced subgraph relation. In Section 5 we also determine, by giving infinite antichains, two bigenic classes that are not, namely the class of (2P 1 + P 2 , P 2 + P 4 )-free graphs, which has unbounded clique-width [11], and the class of (P 1 + P 4 , P 1 + 2P 2 )-free graphs, for which boundedness of the clique-width is unknown (see Fig.…”

Section: Our Resultsmentioning

confidence: 99%

“…In Section 5 we also determine, by giving infinite antichains, two bigenic classes that are not, namely the class of (2P 1 + P 2 , P 2 + P 4 )-free graphs, which has unbounded clique-width [11], and the class of (P 1 + P 4 , P 1 + 2P 2 )-free graphs, for which boundedness of the clique-width is unknown (see Fig. 1 for drawings of the five forbidden induced subgraphs).…”

Section: Our Resultsmentioning

confidence: 99%

“…We divide the proof into several sections, depending on whether or not the graphs under consideration contain certain induced subgraphs or not. We follow the general scheme that Dabrowski, Huang and Paulusma [11] used to prove that this class has bounded clique-width, but we will also need a number of new arguments.…”

Section: A New Well-quasi-ordered Classmentioning

confidence: 99%

“…Proof To prove this, we modify the proof from [11], which shows that this class of graphs has bounded clique-width. Let G be a (2P 1 + P 2 , P 2 + P 3 , K 5 )-free graph containing a C 5 , say on vertices v 1 , v 2 , v 3 , v 4 , v 5 in order.…”

Section: Lemmamentioning

confidence: 99%

“…We prove the following structural result. Proof In order to prove the lemma, we again modify the proof from [11], which shows that this class of graphs has bounded clique-width. Let G be a ( Therefore, Y is a clique.…”

Section: Graphs Containing a Cmentioning

confidence: 99%

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Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes

Dabrowski

Paulusma

2017

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Daligault, Rao and Thomassé asked whether a hereditary class of graphs wellquasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this is not true for classes defined by infinitely many forbidden induced subgraphs. However, in the case of finitely many forbidden induced subgraphs the question remains open and we conjecture that in this case the answer is positive. The conjecture is known to hold for classes of graphs defined by a single forbidden induced subgraph H , as such graphs are well-quasi-ordered and are of bounded clique-width if and only if H is an induced subgraph of P 4 . For bigenic classes of graphs, i.e. ones defined by two forbidden induced subgraphs, there are several open cases in both classifications. In the present paper we obtain a number of new results on well-quasi-orderability of bigenic classes, each of which supports the conjecture.

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Section: Our Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: A New Well-quasi-ordered Classmentioning

confidence: 99%

Section: Lemmamentioning

confidence: 99%

Section: Graphs Containing a Cmentioning

confidence: 99%

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Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes

Dabrowski

Paulusma

2017

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Bounding the mim‐width of hereditary graph classes

Brettell

Horsfield

Munaro

et al. 2021

Journal of Graph Theory

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A large number of NP-hard graph problems are solvable in XP time when parameterized by some width parameter. Hence, when solving problems on special graph classes, it is helpful to know if the graph class under consideration has bounded width. In this paper we consider mim-width, a particularly general width parameter that has a number of algorithmic applications whenever a decomposition is "quickly computable" for the graph class under consideration.We start by extending the toolkit for proving (un)boundedness of mim-width of graph classes. By combining our new techniques with known ones we then initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes, and make a comparison with clique-width, a more restrictive width parameter that has been well studied.We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H1, H2)-free graphs. We identify several general classes of (H1, H2)-free graphs having unbounded clique-width, but bounded mim-width, illustrating the power of mim-width. Moreover, we show that a branch decomposition of constant mim-width can be found in polynomial time, for these classes. Hence, as mentioned, these results have algorithmic implications: when the input is restricted to such a class of (H1, H2)-free graphs, many problems become polynomial-time solvable, including classical problems such as k-Colouring and Independent Set, domination-type problems known as LC-VSVP problems, and distance versions of LC-VSVP problems, to name just a few. We also prove a number of new results showing that, for certain H1 and H2, the class of (H1, H2)-free graphs has unbounded mim-width.Boundedness of clique-width implies boundedness of mim-width. By combining our results, which give both new bounded and unbounded cases for mim-width, with the known bounded cases for clique-width, we present summary theorems of the current state of the art for the boundedness of mim-width for (H1, H2)-free graphs. In particular, we classify the mim-width of (H1, H2)-free graphs for all pairs (H1, H2) with |V (H1)| + |V (H2)| ≤ 8. When H1 and H2 are connected graphs, we classify all pairs (H1, H2) except for one remaining infinite family and a few isolated cases.

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Clique‐width: Harnessing the power of atoms

Dabrowski

Masařík

Novotná

et al. 2023

Journal of Graph Theory

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Many NP‐complete graph problems are polynomial‐time solvable on graph classes of bounded clique‐width. Several of these problems are polynomial‐time solvable on a hereditary graph class if they are so on the atoms (graphs with no clique cut‐set) of . Hence, we initiate a systematic study into boundedness of clique‐width of atoms of hereditary graph classes. A graph is ‐free if is not an induced subgraph of , and it is ‐free if it is both ‐free and ‐free. A class of ‐free graphs has bounded clique‐width if and only if its atoms have this property. This is no longer true for ‐free graphs, as evidenced by one known example. We prove the existence of another such pair and classify the boundedness of clique‐width on ‐free atoms for all but 18 cases.

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Bounding clique-width via perfect graphs

Cited by 12 publications

References 31 publications

Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes

Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes

Bounding the mim‐width of hereditary graph classes

Clique‐width: Harnessing the power of atoms

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