Computing Subset Transversals in H-Free Graphs

Brettell, Nick; Johnson, Matthew; Paesani, Giacomo; Paulusma, Daniël

doi:10.1007/978-3-030-60440-0_15

Cited by 9 publications

(18 citation statements)

References 27 publications

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“…So far, both these problems behave in exactly the same way on H-free graphs as their feedback vertex set counterparts (see [4] and [5]). So, the only open cases for Weighted Subset Odd Cycle Transversal on H-free graphs are the ones where H ∈ {2P 1 +P 3 , P 1 +P 4 , 2P 1 +P 4 } and the only open cases for Subset Odd Cycle Transversal on H-free graphs are the ones where H = sP 1 + P 4 for some s ≥ 1.…”

Section: Discussionmentioning

confidence: 99%

“…However, this is no longer true for the polynomial-time cases: Fomin et al [7] proved NP-completeness of Subset Feedback Vertex Set for split graphs, which form a subclass of 2P 2free graphs. Interestingly, Papadopoulos and Tzimas [13] proved that Weighted Subset Feedback Vertex Set is NP-complete for 5P 1 -free graphs, whereas Brettell et al [4] proved that Subset Feedback Vertex Set can be solved in polynomial time even for (sP 1 + P 3 )-free graphs for every s ≥ 1 [4]. Hence, in contrast to many other transversal problems, the complexities on the weighted and unweighted subset versions do not coincide for H-free graphs.…”

Section: Weighted Feedback Vertex Setmentioning

confidence: 99%

“…The above results leave open a number of unresolved cases for both problems, as identified in [4] and [5], where the following open problems are posed: Open Problem 1 Determine the complexity of Weighted Subset Feedback Vertex Set for H-free graphs if H ∈ {2P 1 + P 3 , P 1 + P 4 , 2P 1 + P 4 }.…”

Section: Weighted Feedback Vertex Setmentioning

confidence: 99%

“…Throughout our proofs we will need to check if some graph F is a solution, that is, if F is a T -forest so does not contain any T -cycles. The following lemma from [4] is easy to prove and shows that we can recognize solutions in linear time. The lemma combines results claimed but not proved in [9,13].…”

Section: Preliminariesmentioning

confidence: 99%

“…We claim that F * is the desired full 3-part solution (for this particular branch). 4 In order to see this, assume that F * − u is not the disjoint union of three complete graphs D 1 , D 2 , D 3 . By construction, this means that one of the following two cases must hold:…”

Section: ⊓ ⊔mentioning

confidence: 99%

See 4 more Smart Citations

Classifying Subset Feedback Vertex Set for $H$-Free Graphs

Paesani¹,

Paulusma²,

Rzążewski³

2022

Preprint

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In the Feedback Vertex Set problem, we aim to find a small set S of vertices in a graph intersecting every cycle. The Subset Feedback Vertex Set problem requires S to intersect only those cycles that include a vertex of some specified set T . We also consider the Weighted Subset Feedback Vertex Set problem, where each vertex u has weight w(u) > 0 and we ask that S has small weight. By combining known NP-hardness results with new polynomial-time results we prove full complexity dichotomies for Subset Feedback Vertex Set and Weighted Subset Feedback Vertex Set for H-free graphs, that is, graphs that do not contain a graph H as an induced subgraph.

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Section: Discussionmentioning

confidence: 99%

Section: Weighted Feedback Vertex Setmentioning

confidence: 99%

Section: Weighted Feedback Vertex Setmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: ⊓ ⊔mentioning

confidence: 99%

See 3 more Smart Citations

Classifying Subset Feedback Vertex Set for $H$-Free Graphs

Paesani¹,

Paulusma²,

Rzążewski³

2022

Preprint

Self Cite

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Computing Weighted Subset Transversals in H-Free Graphs

Brettell

Johnson

Paulusma

2021

Lecture Notes in Computer Science

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For the Odd Cycle Transversal problem, the task is to nd a small set S of vertices in a graph that intersects every cycle of odd length. The Subset Odd Cycle Transversal requires S to intersect only those odd cycles that include a vertex of a distinguished vertex subset T . If we are given weights for the vertices, we ask instead that S has small weight: this is the problem Weighted Subset Odd Cycle Transversal. We prove an almost-complete complexity dichotomy for Weighted Subset Odd Cycle Transversal for graphs that do not contain a graph H as an induced subgraph. Our general approach can also be used for Weighted Subset Feedback Vertex Set, which enables us to generalize a recent result of Papadopoulos and Tzimas.Subset Odd Cycle Transversal [3], which are the special cases with w ≡ 1,

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Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Papadopoulos

Tzimas

2022

Lecture Notes in Computer Science

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Chordal graphs are characterized as the intersection graphs of subtrees in a tree and such a representation is known as the tree model. Restricting the characterization results in well-known subclasses of chordal graphs such as interval graphs or split graphs. A typical example that behaves computationally different in subclasses of chordal graph is the Subset Feedback Vertex Set (SFVS) problem: given a vertex-weighted graph G = (V, E) and a set S ⊆ V , we seek for a vertex set of minimum weight that intersects all cycles containing a vertex of S. SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains NP-complete on split graphs and, consequently, on chordal graphs. Towards a better understanding of the complexity of SFVS on subclasses of chordal graphs, we exploit structural properties of a tree model in order to cope with the hardness of SFVS. Here we consider variants of the leafage that measures the minimum number of leaves in a tree model. We show that SFVS can be solved in polynomial time for every chordal graph with bounded leafage. In particular, given a chordal graph on n vertices with leafage ℓ, we provide an algorithm for SFVS with running time n O(ℓ) , thus improving upon n O(ℓ 2 ) , the running time of the previously known algorithm obtained for graphs with bounded mim-width. We complement our result by showing that SFVS is W[1]-hard parameterized by ℓ. Pushing further our positive result, it is natural to consider a slight generalization of leafage, the vertex leafage, which measures the minimum upper bound on the number of leaves of every subtree in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains NP-complete on undirected path graphs, i.e., chordal graphs having vertex leafage at most two. Lastly, we provide a polynomial-time algorithm for SFVS on rooted path graphs, a proper subclass of undirected path graphs and graphs of mim-width one, which is faster than the previously known algorithm obtained for graphs with bounded mim-width.

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Computing Subset Transversals in H-Free Graphs

Cited by 9 publications

References 27 publications

Classifying Subset Feedback Vertex Set for $H$-Free Graphs

Classifying Subset Feedback Vertex Set for $H$-Free Graphs

Computing Weighted Subset Transversals in H-Free Graphs

Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

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