Rank-Width and Well-Quasi-Ordering

Oum, Sang‐il

doi:10.1137/050629616

Cited by 37 publications

(28 citation statements)

References 12 publications

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“…Corollary 2.6. (Oum [39,40]) For every integer k, there is a finite set C k of graphs such that for every graph G, rwd(G) k if and only if no vertex-minors of G are isomorphic to a graph in C k .…”

Section: Vertex-minor and Well-quasi-orderingmentioning

confidence: 99%

Vertex-minors, monadic second-order logic, and a conjecture by Seese

Courcelle

Oum

2007

Journal of Combinatorial Theory, Series B

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123

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We prove that one can express the vertex-minor relation on finite undirected graphs by formulas of monadic second-order logic (with no edge set quantification) extended with a predicate expressing that a set has even cardinality. We obtain a slight weakening of a conjecture by Seese stating that sets of graphs having a decidable satisfiability problem for monadic second-order logic have bounded clique-width. We also obtain a polynomial-time algorithm to check that the rank-width of a graph is at most k for any fixed k. The proofs use isotropic systems.

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Section: Vertex-minor and Well-quasi-orderingmentioning

confidence: 99%

Vertex-minors, monadic second-order logic, and a conjecture by Seese

Courcelle

Oum

2007

Journal of Combinatorial Theory, Series B

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100

123

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“…However, the cardinality of such an obstruction set can be enormous. It is known that if a class of graphs has bounded rank-width, then the obstruction set is finite [15]. This implies that, for every integer k ≥ 0, the obstruction set for the class of graphs of linear rank-width at most k is finite.…”

Section: Resultsmentioning

confidence: 97%

“…It is known that the obstruction sets for graphs of bounded linear rank-width w.r.t. vertex-minor and pivot-minor containment are finite [14,15]. However, until now none of these obstruction sets were known explicitly.…”

Section: Introductionmentioning

confidence: 99%

Obstructions for linear rank-width at most 1

Adler

Farley

Proskurowski

2014

Discrete Applied Mathematics

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“…Vertex-minors and pivot-minors are graph containment relations introduced by Bouchet [3,4,5,6] while conducting research of circle graphs (intersection graphs of chords in a cycle) and 4-regular Eulerian digraphs. Furthermore, these graph operations have been used for developing theory on rank-width [20,26,27,28,29]. We review these concepts in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors

Choi

Kwon

Oum

2017

Journal of Combinatorial Theory, Series B

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A fan F k is a graph that consists of an induced path on k vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers q and k, every graph with sufficiently large chromatic number contains either a clique of size q or a vertex-minor isomorphic to F k . We also prove that for all positive integers q and k ≥ 3, every graph with sufficiently large chromatic number contains either a clique of size q or a pivot-minor isomorphic to a cycle of length k.

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Rank-Width and Well-Quasi-Ordering

Cited by 37 publications

References 12 publications

Vertex-minors, monadic second-order logic, and a conjecture by Seese

Vertex-minors, monadic second-order logic, and a conjecture by Seese

Obstructions for linear rank-width at most 1

Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors

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