Two forbidden induced subgraphs and well-quasi-ordering

Korpelainen, Nicholas

doi:10.1016/j.disc.2011.04.023

Cited by 39 publications

(74 citation statements)

References 12 publications

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“…We circumvent this by investigating whether these three operations preserve boundedness of a graph parameter called uniformicity. This parameter was introduced by Korpelainen and Lozin [21], who proved that every graph class G of bounded uniformicity is well-quasi-ordered by the so-called labelled induced subgraph relation, which in turn implies that G is well-quasi-ordered by the induced subgraph relation. Korpelainen and Lozin [21] proved that boundedness of uniformicity is preserved by vertex deletion.…”

Section: Our Resultsmentioning

confidence: 99%

“…If two classes are equivalent, then one of them is well-quasi-ordered with respect to the induced subgraph relation if and only if the other one is [21]. Similarly, if two classes are equivalent, then one of them has bounded clique-width if and only if the other one does [14].…”

Section: State Of the Art And Future Workmentioning

confidence: 99%

“…For monogenic classes, the conjecture is true. In this case, the two notions even coincide: a class of graphs defined by a single forbidden induced subgraph H is well-quasi-ordered if and only if it has bounded clique-width if and only if H is an induced subgraph of P 4 (see, for instance, [14,16,21]). …”

Section: Conjecture 1 If a Finitely Defined Class Of Graphs G Is Wellmentioning

confidence: 99%

“…Recently, considerable progress has been made towards answering the latter question for bigenic classes; there are currently seven (non-equivalent) open cases [13]. With respect to well-quasi-orderability of bigenic classes, Korpelainen and Lozin [21] left all but 14 cases open. Since then, Atminas and Lozin [1] proved that the class of (K 3 , P 6 )-free graphs is well-quasi-ordered by the induced subgraph relation and that the class of (2P 1 + P 2 , P 6 )-free graphs is not.…”

Section: Conjecture 1 If a Finitely Defined Class Of Graphs G Is Wellmentioning

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: State Of the Art And Future Workmentioning

confidence: 99%

Section: Conjecture 1 If a Finitely Defined Class Of Graphs G Is Wellmentioning

confidence: 99%

Section: Conjecture 1 If a Finitely Defined Class Of Graphs G Is Wellmentioning

confidence: 99%

See 2 more Smart Citations

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

Dabrowski

Paulusma

2017

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“…For example, it is conjectured in [19] that the P 7 -free bipartite graphs are not wqo under the induced subgraph relation; in [36] it is shown that this is indeed the case, but that the P 6 -free bipartite graphs are wqo. In [37], wqo classes of graphs defined by more than one induced subgraph obstruction are considered.…”

Section: Subgraph Ordermentioning

confidence: 99%

Well quasi-order in combinatorics: embeddings and homomorphisms

Huczynska¹,

Ruškuc²

2015

Surveys in Combinatorics 2015

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The notion of well quasi-order (wqo) from the theory of ordered sets often arises naturally in contexts where one deals with infinite collections of structures which can somehow be compared, and it then represents a useful discriminator between 'tame' and 'wild' such classes. In this article we survey such situations within combinatorics, and attempt to identify promising directions for further research. We argue that these are intimately linked with a more systematic and detailed study of homomorphisms in combinatorics.

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Well‐quasi‐ordering and finite distinguishing number

Atminas

Brignall

2019

Journal of Graph Theory

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Balogh, Bollobás and Weinreich showed that a parameter that has since been termed the distinguishing number can be used to identify a jump in the possible speeds of hereditary classes of graphs at the sequence of Bell numbers.We prove that every hereditary class that lies above the Bell numbers and has finite distinguishing number contains a boundary class for well-quasi-ordering. This means that any such hereditary class which in addition is defined by finitely many minimal forbidden induced subgraphs must contain an infinite antichain. As all hereditary classes below the Bell numbers are well-quasi-ordered, our results complete the answer to the question of well-quasi-ordering for hereditary classes with finite distinguishing number.We also show that the decision procedure of Atminas, Collins, Foniok and Lozin to decide the Bell number (and which now also decides well-quasi-ordering for classes of finite distinguishing number) has run time bounded by an explicit (quadruple exponential) function of the order of the largest minimal forbidden induced subgraph of the class.

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Two forbidden induced subgraphs and well-quasi-ordering

Cited by 39 publications

References 12 publications

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

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

Well quasi-order in combinatorics: embeddings and homomorphisms

Well‐quasi‐ordering and finite distinguishing number

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