Polynomial-Time Algorithms for the Subset Feedback Vertex Set Problem on Interval Graphs and Permutation Graphs

Abstract

Given a vertex-weighted graph G = (V, E) and a set S ⊆ V , a subset feedback vertex set X is a set of the vertices of G such that the graph induced by V \ X has no cycle containing a vertex of S. The Subset Feedback Vertex Set problem takes as input G and S and asks for the subset feedback vertex set of minimum total weight. In contrast to the classical Feedback Vertex Set problem which is obtained from the Subset Feedback Vertex Set problem for S = V , restricted to graph classes the Subset Feedback Vertex Set problem is known to be NP-complete on split graphs and, consequently, on chordal graphs. However as Feedback Vertex Set is polynomially solvable for AT-free graphs, no such result is known for the Subset Feedback Vertex Set problem on any subclass of AT-free graphs. Here we give the first polynomial-time algorithms for the problem on two unrelated subclasses of AT-free graphs: interval graphs and permutation graphs. As a byproduct we show that there exists a polynomial-time algorithm for circular-arc graphs by suitably applying our algorithm for interval graphs. Moreover towards the unknown complexity of the problem for AT-free graphs, we give a polynomial-time algorithm for co-bipartite graphs. Thus we contribute to the first positive results of the Subset Feedback Vertex Set problem when restricted to graph classes for which Feedback Vertex Set is solved in polynomial time.

Computing Subset Transversals in H-Free Graphs

“…If H has a cycle or claw, we use Theorem 1. The cases H = P 4 and H = 2P 2 follow from the corresponding results for permutation graphs [31] and split graphs [16]. The remaining case H ⊆ i sP 1 +P 3 follows from Theorem 6.…”

“…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.…”