Computing Weighted Subset Transversals in H-Free Graphs

Brettell, Nick; Johnson, Matthew; Paulusma, Daniël

doi:10.1007/978-3-030-83508-8_17

Cited by 8 publications

(16 citation statements)

References 17 publications

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“…Moreover it would be interesting to consider the close related problem Subset Odd Cycle Transversal in which the task is to hit all odd S-cycles. Preliminary results indicate that the two problems align on particular hereditary classes of graphs [7,8]. As a byproduct, it is notable that all of our results obtained within this work are still valid for Subset Odd Cycle Transversal, as any induced cycle is an odd induced cycle (triangle) in chordal graphs.…”

Section: Discussionsupporting

confidence: 54%

“…An interesting remark concerning Subset Feedback Vertex Set, is the fact that its unweighted and weighted variants behave computationally different on hereditary graph classes. For example, Subset Feedback Vertex Set is NP-complete on H-free graphs for some fixed graphs H, while its unweighted variant admits polynomial time algorithm on the same class of graphs [8,29]. Thus Subset Feedback Vertex Set is one of the few problems for which its unweighted and weighted variants do not align.…”

Section: Introductionmentioning

confidence: 99%

“…Subset Feedback Vertex Set remains NP-complete on bipartite graphs [34] and planar graphs [16], as a generalization of Feedback Vertex Set. Notable differences between the two later problems regarding their complexity status is the class of split graphs and 4P 1 -free graphs for which Subset Feedback Vertex Set is NP-complete [14,29], as opposed to the Feedback Vertex Set problem [12,32,8]. Inspired by the NP-completeness on chordal graphs, Subset Feedback Vertex Set restricted on (subclasses of) chordal graphs has attracted several researchers to obtain faster, still exponential-time, algorithms [18,30].…”

Section: Introductionmentioning

confidence: 99%

“…On the positive side, Subset Feedback Vertex Set can be solved in polynomial time on restricted graph classes [8,7,28,29]. Among the stated classes, we stress that interval graphs is the only known subclass of chordal graphs for which Subset Feedback Vertex Set is solved in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

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Computing Subset Feedback Vertex Set via Leafage

Papadopoulos¹,

Tzimas²

2021

Preprint

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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 graph G = (V, E) and a set S ⊆ V , SFVS asks for a minimum set of vertices 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 , we provide an algorithm for SFVS with running time n O( ) . Pushing further our positive result, it is natural to consider a slight generalization of leafage, the vertex leafage, which measures the smallest number among the maximum number of leaves of all subtrees 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., graphs having vertex leafage at most two. Moreover, we strengthen previously-known polynomial-time algorithm for SFVS on directed path graphs that form a proper subclass of undirected path graphs and graphs of mim-width one.

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Section: Discussionsupporting

confidence: 54%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computing Subset Feedback Vertex Set via Leafage

Papadopoulos¹,

Tzimas²

2021

Preprint

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“…Table 1 still shows some missing cases for each of the three problems, in particular: Is each of the three problems is polynomial-time solvable for (P + P 4 )-free graphs? Interestingly, this case is also open for two well-known generalizations of Feedback Vertex Set, namely Subset Feedback Vertex Set and Weighted Subset Feedback Vertex Set (see [7,8]). The main obstacle is that we know no polynomial-time algorithm for finding a maximum induced disjoint union of stars in a (P 1 + P 4 )-free graph; note that such a subgraph could be a potential optimal solution for each of the three problems.…”

Section: Discussionmentioning

confidence: 99%

Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs

Paesani¹,

Paulusma²,

Rzążewski³

2021

Preprint

Self Cite

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We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. In particular, we prove that both problems are polynomial-time solvable for sP3-free graphs for every integer s ≥ 1. Our results show that both problems exhibit the same behaviour on H-free graphs (subject to some open cases). This is in part explained by a new general algorithm we design for finding in a graph G a largest induced subgraph whose blocks belong to some finite class C of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on H-free graphs. ACM Subject Classification Mathematics of computing → Graph algorithmsKeywords and phrases Feedback vertex set, even cycle transversal, odd cactus, forest, block

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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 Weighted Subset Transversals in H-Free Graphs

Cited by 8 publications

References 17 publications

Computing Subset Feedback Vertex Set via Leafage

Computing Subset Feedback Vertex Set via Leafage

Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs

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

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