A more accurate view of the Flat Wall Theorem

Sau, Ignasi; Stamoulis, Giannos; Thilikos, Dimitrios M.

doi:10.48550/arxiv.2102.06463

Cited by 2 publications

(37 citation statements)

References 24 publications

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“…In a nutshell, our approach is based on introducing a robust combinatorial framework for finding irrelevant vertices. In fact, what we find is annotation-irrelevant flat territories, building on our previous recent work [7,7,46,[108][109][110][111], which is formulated with enough generality so as to allow for the application of powerful tools such as Gaifman's locality theorem (see Proposition 10) or a variant of Courcelle's theorem on boundaried graphs (see Proposition 26).…”

Section: General Scheme Of the Algorithmmentioning

confidence: 76%

“…Techniques. The algorithm and the proofs of Theorem 5 use as departure point core techniques from the proofs of Propositions 1, 2, and 3 such as Courcelle's theorem for dealing with CMSOLsentences, the use of Gaifman's theorem for dealing with FOL-sentences, and an extended version of the irrelevant vertex technique, introduced by Robertson and Seymour in [105], along with some suitable version of the Flat Wall theorem appeared recently in [80,110] (see also [7,108,109,111]). The algorithm produces equivalent and gradually "strictly simpler" instances of an annotated version of the problem.…”

Section: Our Resultsmentioning

confidence: 99%

“…We conclude the paper with Section 13 by mentioning the limitations of our approach, possible extensions, and open research directions. In Appendix A we present the flat wall framework that we use in this paper, which was introduced in [110]. Also, in Appendix B, we provide all the details of the proofs of Section 10 that are omitted.…”

Section: Our Resultsmentioning

confidence: 99%

“…Hence, flat walls are only "locally non-planar", and after removing apices we can apply useful locality arguments, in the sense that two vertices that are in "distant" flaps should also be "distant" in the whole graph without the apices. One of the most celebrated results in the theory of Graph Minors by Robertson and Seymour [105,106], known as the Flat Wall theorem (see Proposition 59 for a variant recently proved in [80,110]), informally states that graphs of large treewidth contain either a large clique minor or a large flat wall. In this article we use the framework recently introduced in [110] that provides a more accurate view of some previously defined notions concerning flat walls, particularly in [80].…”

Section: General Scheme Of the Algorithmmentioning

confidence: 99%

“…One of the most celebrated results in the theory of Graph Minors by Robertson and Seymour [105,106], known as the Flat Wall theorem (see Proposition 59 for a variant recently proved in [80,110]), informally states that graphs of large treewidth contain either a large clique minor or a large flat wall. In this article we use the framework recently introduced in [110] that provides a more accurate view of some previously defined notions concerning flat walls, particularly in [80]. We provide these precise definitions in Subsection A.…”

Section: General Scheme Of the Algorithmmentioning

confidence: 99%

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A Compound Logic for Modification Problems: Big Kingdoms Fall from Within

Fomin¹,

Golovach²,

Sau³

et al. 2021

Preprint

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We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. We propose a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the target sentence, defining some property of the resulting graph. In our framework, modulator sentences are in monadic second-order logic and have models of bounded treewidth, while target sentences express firstorder logic properties along with minor-exclusion. Our logic captures problems that are not definable in first-order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modeling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or G-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model checking can be done in quadratic time. This algorithmic meta-theorem encompasses, unifies, and extends all known meta-algorithmic results on minor-closed graph classes. Moreover, all derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique. It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle's theorem does not apply. Our insight is that, for problems definable in our logic, such "big kingdoms" can be "corrupted internally": way deep inside instances of big treewidth, we can find "irrelevant territories" that can be safely discarded towards creating simpler equivalent instances. To prove this, we combine novel combinatorial results related to the Flat Wall theorem along with elements of the proof of Courcelle's theorem and Gaifman's theorem.

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Section: General Scheme Of the Algorithmmentioning

confidence: 76%

Section: Our Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: General Scheme Of the Algorithmmentioning

confidence: 99%

Section: General Scheme Of the Algorithmmentioning

confidence: 99%

See 3 more Smart Citations

A Compound Logic for Modification Problems: Big Kingdoms Fall from Within

Fomin¹,

Golovach²,

Sau³

et al. 2021

Preprint

Self Cite

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k -apices of Minor-closed Graph Classes. II. Parameterized Algorithms

Stamoulis

Thilikos

2022

ACM Trans. Algorithms

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Let \(\mathcal {G} \) be a minor-closed graph class. We say that a graph \(G \) is a \(k \) -apex of \(\mathcal {G} \) if \(G \) contains a set \(S \) of at most \(k \) vertices such that \(G\setminus S \) belongs to \(\mathcal {G} \) . We denote by \(\mathcal {A}_k (\mathcal {G}) \) the set of all graphs that are \(k \) -apices of \(\mathcal {G}. \) In the first paper of this series we obtained upper bounds on the size of the graphs in the minor-obstruction set of \(\mathcal {A}_k (\mathcal {G}) \) , i.e., the minor-minimal set of graphs not belonging to \(\mathcal {A}_k (\mathcal {G}). \) In this article we provide an algorithm that, given a graph \(G \) on \(n \) vertices, runs in time \(2^{\textsf {poly}(k)}\cdot n^3 \) and either returns a set \(S \) certifying that \(G \in \mathcal {A}_k (\mathcal {G}) \) , or reports that \(G \notin \mathcal {A}_k (\mathcal {G}) \) . Here \(\textsf {poly} \) is a polynomial function whose degree depends on the maximum size of a minor-obstruction of \(\mathcal {G}. \) In the special case where \(\mathcal {G} \) excludes some apex graph as a minor, we give an alternative algorithm running in \(2^{\textsf {poly}(k)}\cdot n^2 \) -time.

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A more accurate view of the Flat Wall Theorem

Cited by 2 publications

References 24 publications

A Compound Logic for Modification Problems: Big Kingdoms Fall from Within

A Compound Logic for Modification Problems: Big Kingdoms Fall from Within

k -apices of Minor-closed Graph Classes. II. Parameterized Algorithms

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