In absence of long chordless cycles, large tree-width becomes a local phenomenon

Weißauer, Daniel

doi:10.1016/j.jctb.2019.04.004

Cited by 14 publications

(8 citation statements)

References 20 publications

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“…In particular, for a fixed value of k, their approach leads to a linear-time algorithm for the k-Clique and List k-Coloring problems in any such graph class. From the structural point of view, identifying new (tw, ω)-bounded graph classes directly addresses a recent question of Weißauer [62] asking for which classes can we force large cliques by assuming large treewidth. Weißauer distinguishes graph parameters as being either global or local (see [62] for precise definitions).…”

Section: Background and Motivationmentioning

confidence: 99%

“…From the structural point of view, identifying new (tw, ω)-bounded graph classes directly addresses a recent question of Weißauer [62] asking for which classes can we force large cliques by assuming large treewidth. Weißauer distinguishes graph parameters as being either global or local (see [62] for precise definitions). In this terminology, (tw, ω)-boundedness of a graph class is a sufficient condition for treewidth to become a local parameter.…”

Section: Background and Motivationmentioning

confidence: 99%

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Treewidth versus clique number. I. Graph classes with a forbidden structure

Dallard,

Milanič,

Štorgel

2020

Preprint

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Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes in which this condition is also sufficient, which we call (tw, ω)-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and k-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, induced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs H for which the class of graphs excluding H is (tw, ω)-bounded. Our results imply that the class of 1-perfectly orientable graphs is (tw, ω)-bounded, answering a question of Brešar, Hartinger, Kos, and Milanič from 2018. We also reveal some further algorithmic implications of (tw, ω)-boundedness related to list k-coloring and clique problems.

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Section: Background and Motivationmentioning

confidence: 99%

Section: Background and Motivationmentioning

confidence: 99%

Treewidth versus clique number. I. Graph classes with a forbidden structure

Dallard,

Milanič,

Štorgel

2020

Preprint

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“…This result for bounded degree graphs was obtained by reducing it to graphs of bounded chordality (the length of a longest chordless cycle), because bounded chordality together with bounded degree imply bounded tree-width [2]. In [9], the result for bounded chordality was extended from graphs of bounded degree to graphs of bounded biclique number by means of Theorem 1 and the following result proved in [5]. Theorem 3.…”

Section: Theorem 2 [6]mentioning

confidence: 99%

Tree-width dichotomy

Lozin,

Razgon

2020

Preprint

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“…It is a general trend that structural properties regarding induced and non-induced subgraphs tend to coincide on weakly sparse classes. This can particularly be seen in work on induced subdivisions [9,3] and width parameters [6,15]. See the relevant sections of [11] and [13] for summaries of results in this direction.…”

Section: Conjecture 3 (Thomassen [14]mentioning

confidence: 97%