Bipartite induced subgraphs and well-quasi-ordering

Korpelainen, Nicholas

doi:10.1002/jgt.20528

Cited by 23 publications

(35 citation statements)

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“…• (P 7 , Sun 1 )-free bipartite graphs. Well-quasi-orderability of this class was also shown in [10]. With no extra work this result can be extended to the labelled induced subgraph relation, because, as was shown in [10], every connected graph in this class is C 4 -free and (P 7 , C 4 )-free bipartite graphs contain no P 9 as a (not necessarily induced) subgraph.…”

Section: Subclasses Of Bipartite Graphsmentioning

confidence: 74%

“…In [10], it was shown that graphs in this class are k-letter graphs. Therefore, they are well-quasi-ordered by the labelled induced subgraph relation.…”

Section: Subclasses Of Bipartite Graphsmentioning

confidence: 98%

“…There are many exotic antichains with respect to this relation and only few examples of graph classes which are well-quasi-ordered under this relation are known in the literature (see e.g. [5,10,11]). Most of these examples deal with classes defined by finitely many forbidden induced subgraphs.…”

Section: Introductionmentioning

confidence: 99%

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Labelled Induced Subgraphs and Well-Quasi-Ordering

Atminas

2014

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It is known that the set of all simple graphs is not well-quasi-ordered by the induced subgraph relation, i.e. it contains infinite antichains (sets of incomparable elements) with respect to this relation. However, some particular graph classes are well-quasi-ordered by induced subgraphs. Moreover, some of them are well-quasi-ordered by a stronger relation called labelled induced subgraphs. In this paper, we conjecture that a hereditary class X which is well-quasi-ordered by the induced subgraph relation is also well-quasi-ordered by the labelled induced subgraph relation if and only if X is defined by finitely many minimal forbidden induced subgraphs. We verify this conjecture for a variety of hereditary classes that are known to be well-quasi-ordered by induced subgraphs and prove a number of new results supporting the conjecture.

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Section: Subclasses Of Bipartite Graphsmentioning

confidence: 74%

“…In [10], it was shown that graphs in this class are k-letter graphs. Therefore, they are well-quasi-ordered by the labelled induced subgraph relation.…”

Section: Subclasses Of Bipartite Graphsmentioning

confidence: 98%

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Labelled Induced Subgraphs and Well-Quasi-Ordering

Atminas

2014

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“…Special classes, such as the set of bipartite graphs, have also been investigated under induced subgraph order. 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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“…Similarly, a bit later Ding proved in [5] an analogous result for the subgraph relation. Other authors also considered this problem (see for instance [2,[11][12][13]). In this paper, we provide an answer to the same question for the induced minor relation, which we denote ≤ im .…”

Section: Introductionmentioning

confidence: 99%

Induced minors and well-quasi-ordering

Błasiok

Kamiński

Raymond

et al. 2015

Electronic Notes in Discrete Mathematics

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A graph H is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be Hinduced minor-free. Robin Thomas showed that K 4 -induced minor-free graphs are well-quasi-ordered by induced minors [Graphs without K 4 and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3): [240][241][242][243][244][245][246][247] 1985].We provide a dichotomy theorem for H-induced minor-free graphs and show that the class of H-induced minor-free graphs is well-quasi-ordered by induced minors if and only if H is an induced minor of the Gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices. To this end we prove two decomposition theorems which are of independent interest.Similar dichotomy results were previously given for subgraphs by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5): [489][490][491][492][493][494][495][496][497][498][499][500][501][502] 1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and well-quasi

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Bipartite induced subgraphs and well-quasi-ordering

Cited by 23 publications

References 8 publications

Labelled Induced Subgraphs and Well-Quasi-Ordering

Labelled Induced Subgraphs and Well-Quasi-Ordering

Well quasi-order in combinatorics: embeddings and homomorphisms

Induced minors and well-quasi-ordering

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