Hitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable

Golovach, Petr A.; Stamoulis, Giannos; Thilikos, Dimitrios M.

doi:10.1137/1.9781611975994.56

Cited by 7 publications

(10 citation statements)

References 25 publications

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“…• In order to reroute (colored levelings of) topological minor models, it will be helpful to use railed annuli, a structure introduced in [29] that occurs as a subgraph inside a flat wall (Proposition 12) and that has the following nice property, recently proved in [26] (Proposition 13): if a railed annulus is large enough compared to h, every topological minor model of a graph on at most h vertices traversing it can be rerouted so that the branch vertices are preserved and such that, more importantly, the intersection of the new model with a large prescribed part of the railed annulus is confined, in the sense that it is only allowed to use a well-defined set of paths in that part, which does not depend on the original model.…”

Section: Overview Of the Algorithmmentioning

confidence: 99%

“…According to the proof of Proposition 13 in [26], the function f 6 emerges by the following known result, known as the Unique Linkage Theorem. Proposition 29 ([33, 45]).…”

Section: Upper Bounds On the Constants Depending On The Excluded Minorsmentioning

confidence: 99%

“…Another reason is that in our rerouting procedure, in order to find an irrelevant vertex (Theorem 16), we may find a different topological minor model that corresponds to the same minor. Nevertheless, we think that this latter difficulty can be overcome for planar graphs -or even minor-free graphs-by making use of the rerouting potential of Proposition 13, as this is done in [26] for planar graphs. Figure 26: Classification of the complexity of {H}-M-Deletion for all connected simple graphs H with 2 ≤ |V (H)| ≤ 5: for the nine graphs on the left (resp.…”

Section: Further Researchmentioning

confidence: 99%

See 2 more Smart Citations

A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary

Baste

Thilikos

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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For a fixed connected graph H, the {H}-M-Deletion problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G. It is known that this problem can be solved in time f (tw) · n O(1) , where tw is the treewidth of G. We determine the asymptotically optimal function f (tw), for each possible choice of H. Namely, we prove that, under the ETH, f (tw) = 2 Θ(tw) if H is a contraction of the chair or the banner, and f (tw) = 2 Θ(tw · log tw) otherwise. Prior to this work, such a complete characterization was only known when H is a planar graph with at most five vertices. For the upper bounds, we present an algorithm in time 2 Θ(tw · log tw) · n O(1) for the more general problem where all minor models of connected graphs in a finite family F need to be hit. We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement. In particular, this algorithm vastly generalizes a result of Jansen et al. [SODA 2014] for the particular case F = {K 5 , K 3,3 }. For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in [IPEC 2018], which uses the framework introduced by Lokshtanov et al. [SODA 2011] to obtain superexponential lower bounds.

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Section: Overview Of the Algorithmmentioning

confidence: 99%

“…According to the proof of Proposition 13 in [26], the function f 6 emerges by the following known result, known as the Unique Linkage Theorem. Proposition 29 ([33, 45]).…”

Section: Upper Bounds On the Constants Depending On The Excluded Minorsmentioning

confidence: 99%

Section: Further Researchmentioning

confidence: 99%

See 1 more Smart Citation

A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary

Baste

Thilikos

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The following result is the core result of [111] (which, in turn, is based on the results of [7] and [64]). We refer the reader to Subsection A.4 for a definition of a flatness pair.…”

Section: Proposition 28mentioning

confidence: 99%

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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“…Very recently Golovach et al [38] proved that the special case of TM-Deletion where the input graph G is required to be planar is FPT. They conjecture that this is the case for also for general input graphs G. Our main result is a proof of this conjecture.…”

Section: Topological Minor Deletion (Tm-deletion)mentioning

confidence: 99%

Hitting Topological Minors is FPT

Fomin¹,

Lokshtanov²,

Panolan³

et al. 2019

Preprint

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In the Topological Minor Deletion (TM-Deletion) problem input consists of an undirected graph G, a family of undirected graphs F and an integer k. The task is to determine whether G contains a set of vertices S of size at most k, such that the graph G \ S obtained from G by removing the vertices of S, contains no graph from F as a topological minor. We give an algorithm for TM-Deletion with running time f (h , k) • |V (G)| 4 . Here h is the maximum size of a graph in F and f is a computable function of h and k. This is the first fixed parameter tractable algorithm (FPT) for the problem. In fact, even for the restricted case of planar inputs the first FPT algorithm was found only recently by Golovach et al. [SODA 2020]. For this case we improve upon the algorithm of Golovach et al. [SODA 2020] by designing an FPT algorithm with explicit dependence on k and h .

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Hitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable

Cited by 7 publications

References 25 publications

A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary

A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary

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

Hitting Topological Minors is FPT

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