Computing subset transversals in H-free graphs

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

doi:10.1016/j.tcs.2021.12.010

Cited by 4 publications

(9 citation statements)

References 39 publications

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“…We now mention the polynomial-time results on H-free graphs for the unweighted subset variants of the problems (which do not imply anything for the weighted subset versions). It is known that Subset Odd Cycle Transversal is polynomial-time solvable on (sP 1 + P 3 )free graphs for every integer s ≥ 0 [6] and on P 4 -free graphs [6]. In Section 6 we show that the latter result can be generalized to the weighted variant in a straightforward way.…”

Section: Past Resultsmentioning

confidence: 86%

“…There is no linear forest H for which Feedback Vertex Set on H-free graphs is known to be NP-complete, but for Odd Cycle Transversal we can take H = P 2 + P 5 or H = P 6 , as the latter problem is NP-complete even for (P 2 + P 5 , P 6 )-free graphs [12]. It is known that Subset Feedback Vertex Set [14] and Subset Odd Cycle Transversal [6], which are the special cases with w ≡ 1, are NP-complete for 2P 2 -free graphs; in fact, these results were proved for split graphs which form a proper subclass of 2P 2 -free graphs. Papadopoulos and Tzimas [28] proved the following interesting dichotomy, which motivated our research.…”

Section: Past Resultsmentioning

confidence: 99%

“…We make use of an algorithm that decides the problem on P 4 -free graphs. It can be shown that Weighted Subset Odd Cycle Transversal is polynomial-time solvable for P 4free graphs by an obvious adaptation of the proof of the unweighted variant of Subset Odd Cycle Transversal from [6]. However, we remark that, as is the case for Weighted Subset Feedback Vertex Set discussed in Section 1, the result also follows from a metatheorem of Courcelle et al [11] that shows that on graph classes of bounded clique-width, certain optimization problems have linear time algorithms.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…We need a result on this problem for P 4 -free graphs, which can be proven in the same way as the unweighted variant of Subset Vertex Cover in [6]. Alternatively, the property that a subset of vertices S is an T -vertex cover in a graph G = (V, E) for some given set T ⊆ V can be expressed in MSO 1 monadic second-order logic (with S as the only free monadic variable) and we can use the meta-theorem of Courcelle et al [11] again.…”

Section: Weighted Subset Vertex Covermentioning

confidence: 99%

“…In the remaining case H is a linear forest. If H contains an induced 2P 2 , then we use a result of [6], which states that Subset Odd Cycle Transversal is NP-complete for split graphs, or equivalently, (C 4 , C 5 , 2P 2 )-free graphs. If H contains an induced 5P 1 , then we use Theorem 6.…”

Section: The Proof Of Theoremmentioning

confidence: 99%

See 4 more Smart Citations

Computing Weighted Subset Odd Cycle Transversals in H-free graphs

Brettell

Johnson

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2022

Journal of Computer and System Sciences

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Section: Past Resultsmentioning

confidence: 86%

Section: Past Resultsmentioning

confidence: 99%

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Weighted Subset Vertex Covermentioning

confidence: 99%

Section: The Proof Of Theoremmentioning

confidence: 99%

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Computing Weighted Subset Odd Cycle Transversals in H-free graphs

Brettell

Johnson

Paulusma

2022

Journal of Computer and System Sciences

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Computing Subset Vertex Covers in H-Free Graphs

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

Papadopoulos

Tzimas

2023

Algorithmica

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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 of a problem that does not behave computationally the same in all subclasses of chordal graphs is the Subset Feedback Vertex Set (SFVS) problem: given a vertex-weighted graph $$G=(V,E)$$ G = ( V , E ) and a set $$S\subseteq V$$ 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 the leafage, which 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 $$\ell $$ ℓ , we provide an algorithm for solving SFVS with running time $$n^{O(\ell )}$$ n O ( ℓ ) , thus improving upon $$n^{O(\ell ^2)}$$ n O ( ℓ 2 ) , which is the running time of an approach that utilizes the previously known algorithm for graphs with bounded mim-width. We complement our result by showing that SFVS is w[1]-hard parameterized by $$\ell $$ ℓ . Pushing further our positive result, it is natural to also consider 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 solving SFVS on rooted path graphs, a proper subclass of undirected path graphs and graphs with mim-width one, which is faster than the approach of constructing a graph decomposition of mim-width one and applying the previously known algorithm for graphs with bounded mim-width.

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Computing subset transversals in H-free graphs

Cited by 4 publications

References 39 publications

Computing Weighted Subset Odd Cycle Transversals in H-free graphs

Computing Weighted Subset Odd Cycle Transversals in H-free graphs

Computing Subset Vertex Covers in H-Free Graphs

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

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