Subset feedback vertex set on graphs of bounded independent set size

Abstract

The (Weighted) Subset Feedback Vertex Set problem is a generalization of the classical Feedback Vertex Set problem and asks for a vertex set of minimum (weighted) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although the two problems exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of a structural graph parameter. Here we consider graphs of bounded independent set number for which it is known that Weighted Feedback Vertex Set is solved in polynomial time. We provide a dichotomy result with respect to the size of a maximum independent set. In particular we show that Weighted Subset Feedback Vertex Set can be solved in polynomial time for graphs of independent set number at most three, whereas we prove that the problem remains NP-hard for graphs of independent set number four. Moreover, we show that the (unweighted) Subset Feedback Vertex Set problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time n O(d) , where d is the size of a maximum independent set of the input graph. To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size. Based on our findings for Subset Feedback Vertex Set, we settle the complexity of Node Multiway Cut, a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals, as well as its variants where nodes are weighted and/or the terminals are deletable, for every value of the given independent set number.

Computing Subset Transversals in H-Free Graphs

“…We will use the following easy lemma, which proves that T -forests and T -bipartite graphs can be recognized in polynomial time. It combines results claimed but not proved in [27,32].…”

“…This situation changes for Subset Feedback Vertex Set which is, unlike Feedback Vertex Set, NP-complete for split graphs (that is, (2P 2 , C 4 , C 5 )-free graphs), as shown by Fomin et al [16]. Papadopoulos and Tzimas [31,32] proved that Subset Feedback Vertex Set is polynomial-time solvable for sP 1 -free graphs for any s ≥ 1, co-bipartite graphs, interval graphs and permutation graphs, and thus P 4 -free graphs. Some of these results were generalized by Bergougnoux et al [2], who solved an open problem of Jake et al [22] by giving an n O(w 2 ) -time algorithm for Subset Feedback Vertex Set given a graph and a decomposition of this graph of mim-width w. This does not lead to new results for H-free graphs: a class of H-free graphs has bounded mim-width if and only if H ⊆ i P 4 [7].…”

Mim-Width II. The Feedback Vertex Set Problem

“…This question was also posed recently by Papadopoulos and Tzimas who gave an XP-time algorithm for Subset Feedback Vertex Set parameterized by the size of an independent set in the input graph [29]. Moreover, they also showed in earlier work that Subset Feedback Vertex Set is polynomial-time solvable on Permutation and Interval graphs [28], both classes of linear mim-width 1.…”

On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest

“…Both Subset Feedback Vertex Set and Subset Odd Cycle Transversal are NP-complete for H-free graphs if H contains a cycle or claw, due to the aforementioned NP-completeness for the original problems. Moreover, Subset Feedback Vertex Set is polynomial-time solvable for sP 1 -free graphs [28] and for permutation graphs [27], and thus for P 4 -free graphs, but NP-complete for split graphs [13], or equivalently, (C 4 , C 5 , 2P 2 )-free graphs, and thus for P 5 -free graphs. 6 It would be interesting to obtain full complexity dichotomies for Subset Feedback Vertex Set and Subset Odd Cycle Transversal for H-free graphs.…”