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

Dabrowski, Konrad K.; Paulusma, Daniël

doi:10.1007/s11083-017-9430-7

Cited by 3 publications

(5 citation statements)

References 28 publications

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“…The class of (P 1 + P 4 , P 2 + P 3 )-free graphs is the only bigenic graph class left for which Conjecture 1 still needs to be verified; see [10] for details of this claim (which can also be deduced from Theorems 6 and 7 below).…”

Section: ⊓ ⊔mentioning

confidence: 98%

“…Lemma 4. [10] The following operations preserve well-quasi-orderability by the labelled induced subgraph relation:…”

Section: Well-quasi-orderabilitymentioning

confidence: 99%

“…The class of (P 1 + P 4 , P 2 +P 3 )-free graphs is (up to an equivalence relation 3 ) one of the six remaining bigenic graph classes for which well-quasi-orderability is still open and one of the six bigenic graph classes for which boundedness of clique-width is still open. We refer to [10] for details of the first claim and to [3] for details of the second claim. To make our paper self-contained we recall two theorems from these papers, which sum up our current knowledge on bigenic graph classes (including the new results of this paper).…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Such graph classes are said to be bigenic or (H 1 , H 2 )-free graph classes. In this case Conjecture 1 is also known to be true except for two stubborn open cases, namely (H 1 , H 2 ) = (K 3 , P 2 + P 4 ) and (H 1 , H 2 ) = (P 1 + P 4 , P 2 + P 3 ); see [10].…”

Section: Introductionmentioning

confidence: 98%

“…[10] The following operations preserve well-quasi-orderability by the labelled induced subgraph relation:(i) Subgraph complementation, (ii) Bipartite complementation and (iii) Vertex deletion. Lemma 5 ([2]).…”

mentioning

confidence: 99%

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Clique-Width and Well-Quasi-Ordering of Triangle-Free Graph Classes

Dabrowski

Paulusma

2017

Graph-Theoretic Concepts in Computer Science

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Abstract. Daligault, Rao and Thomassé asked whether every hereditary graph class that is well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev (JCTB 2017+) gave a negative answer to this question, but their counterexample is a class that can only be characterised by infinitely many forbidden induced subgraphs. This raises the issue of whether the question has a positive answer for finitely defined hereditary graph classes. Apart from two stubborn cases, this has been confirmed when at most two induced subgraphs H1, H2 are forbidden. We confirm it for one of the two stubborn cases, namely for the (H1, H2) = (triangle, P2 + P4) case, by proving that the class of (triangle, P2 + P4)-free graphs has bounded clique-width and is well-quasi-ordered. Our technique is based on a special decomposition of 3-partite graphs. We also use this technique to prove that the class of (triangle, P1 + P5)-free graphs, which is known to have bounded cliquewidth, is well-quasi-ordered. Our results enable us to complete the classification of graphs H for which the class of (triangle, H)-free graphs is well-quasi-ordered.

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Section: ⊓ ⊔mentioning

confidence: 98%

“…Lemma 4. [10] The following operations preserve well-quasi-orderability by the labelled induced subgraph relation:…”

Section: Well-quasi-orderabilitymentioning

confidence: 99%

Section: ⊓ ⊔mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

mentioning

confidence: 99%

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Clique-Width and Well-Quasi-Ordering of Triangle-Free Graph Classes

Dabrowski

Paulusma

2017

Graph-Theoretic Concepts in Computer Science

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Clique-width and well-quasi-ordering of triangle-free graph classes

Dabrowski

Paulusma

2020

Journal of Computer and System Sciences

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Clique-Width for Graph Classes Closed under Complementation

Blanché¹,

Dabrowski²,

Johnson³

et al. 2020

SIAM J. Discrete Math.

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Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set H of forbidden induced subgraphs. We initiate a systematic study into the boundedness of cliquewidth of hereditary graph classes closed under complementation. First, we extend the known classification for the |H| = 1 case by classifying the boundedness of clique-width for every set H of self-complementary graphs. We then completely settle the |H| = 2 case. In particular, we determine one new class of (H, H)-free graphs of bounded clique-width (as a side effect, this leaves only six classes of (H 1 , H 2)-free graphs, for which it is not known whether their cliquewidth is bounded). Once we have obtained the classification of the |H| = 2 case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for a set F of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for ({H, H} ∪ F)-free graphs coincides with the one for the |H| = 2 case if and only if F does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are P 1 and P 4 , and P 4-free graphs have clique-width at most 2). Finally, we discuss the consequences of our results for Colouring.

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

Cited by 3 publications

References 28 publications

Clique-Width and Well-Quasi-Ordering of Triangle-Free Graph Classes

Clique-Width and Well-Quasi-Ordering of Triangle-Free Graph Classes

Clique-width and well-quasi-ordering of triangle-free graph classes

Clique-Width for Graph Classes Closed under Complementation

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