Vertex deletion parameterized by elimination distance and even less

Jansen, Bart M. P.; Kroon, Jari J. H. de; Włodarczyk, Michał

doi:10.1145/3406325.3451068

Cited by 9 publications

(5 citation statements)

References 71 publications

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“…The state of the art for Chordal Vertex Deletion (ChVD) is the running time 2 O(tw 2 ) n O (1) , which follows from a more general result for a hybrid graph measure H-treewidth, where H = chordal [40]. We improve the dependency on treewidth to single-exponential.…”

Section: Our Resultsmentioning

confidence: 93%

“…The concept of representative families, which plays an important role in our algorithm for ChVD, has found applications outside the context of treewidth as well [62,63]. Our other tool, boundaried graphs, has revealed fruitful insights for various graph classes [22,40,64].…”

Section: Related Workmentioning

confidence: 99%

“…Jansen et al [40] showed that any chordal graph H with a boundary of size k can be condensed to a graph H ′ on O(k) vertices that exhibits the same behavior in terms of gluing. More precisely, the gluing product of H with any graph J is chordal if and only if the gluing product of H ′ with J is chordal.…”

Section: Chordal Vertex Deletionmentioning

confidence: 99%

“…[73, Def. 5.55]) For a graph G and a vertex set X ⊆ V (G) let the graph Condense(G, X) be obtained from G by contracting the connected connected components of G − X into single vertices and then removing those of them which are simplicial.…”

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confidence: 99%

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Tight Bounds for Chordal/Interval Vertex Deletion Parameterized by Treewidth

Wlodarczyk

2023

Preprint

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In Chordal/Interval Vertex Deletion we ask how many vertices one needs to remove from a graph to make it chordal (respectively: interval). We study these problems under the parameterization by treewidth (tw) of the input graph G. On the one hand, we present an algorithm for Chordal Vertex Deletion with running time 2^O(tw)*|V(G)|, improving upon the running time 2^O(tw^2)*|V(G)|^O(1) by Jansen, de Kroon, and Wlodarczyk (STOC'21). When a tree decomposition of width tw is given, then the base of the exponent equals 2^(omega-1)*3+1. Our algorithm is based on a novel link between chordal graphs and graphic matroids, which allows us to employ the framework of representative families. On the other hand, we prove that the known 2^O(tw log tw)*|V(G)|-time algorithm for Interval Vertex Deletion cannot be improved assuming Exponential Time Hypothesis.

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Section: Our Resultsmentioning

confidence: 93%

Section: Related Workmentioning

confidence: 99%

Section: Chordal Vertex Deletionmentioning

confidence: 99%

mentioning

confidence: 99%

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Tight Bounds for Chordal/Interval Vertex Deletion Parameterized by Treewidth

Wlodarczyk

2023

Preprint

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“…An interesting open problem is whether similar polynomial kernels can be obtained for other F-Minor Free Deletion problems. Regarding the field of fixed parameter tractable algorithms, it was recently shown [19] that for any set F of connected graphs, F-Minor Free Deletion admits an FPT algorithm when parameterized by the elimination distance to an F-minor free graph (or even H-treewidth when H is the class of F-minor free graphs). This generalizes known FPT algorithms for the natural parameterization by solution size.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Kernelization for Feedback Vertex Set via Elimination Distance to a Forest

Dekker

Jansen

2022

Graph-Theoretic Concepts in Computer Science

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We study efficient preprocessing for the undirected Feedback Vertex Set problem, a fundamental problem in graph theory which asks for a minimum-sized vertex set whose removal yields an acyclic graph. More precisely, we aim to determine for which parameterizations this problem admits a polynomial kernel. While a characterization is known for the related Vertex Cover problem based on the recently introduced notion of bridge-depth, it remained an open problem whether this could be generalized to Feedback Vertex Set. The answer turns out to be negative; the existence of polynomial kernels for structural parameterizations for Feedback Vertex Set is governed by the elimination distance to a forest. Under the standard assumption $$\textrm{NP}\not \subseteq \textrm{coNP}/\textrm{poly}$$, we prove that for any minor-closed graph class $$\mathcal {G}$$, Feedback Vertex Set parameterized by the size of a modulator to $$\mathcal {G}$$ has a polynomial kernel if and only if $$\mathcal {G}$$ has bounded elimination distance to a forest. This captures and generalizes all existing kernels for structural parameterizations of the Feedback Vertex Set problem.

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