Simultaneous Feedback Vertex Set

Abstract

For a family F of graphs, a graph G , and a positive integer k , the F -D eletion problem asks whether we can delete at most k vertices from G to obtain a graph in F . F -D eletion generalizes many classical graph problems such as V ertex C over , F eedback V ertex S et , and O dd C ycle T ransversal . For an integer α ≥ 1, an n -vertex (multi) graph G = ( V , ⋃ i=1 α E i ), where the edge set of G is partitioned into α color classes, is called an α-edge-colored (multi) graph. A natural extension of the F -D eletion problem to edge-colored graphs is the S imultaneous F -D eletion problem. In the latter problem, we are given an α-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph G i − S , where G i = ( V , E i ) and 1 ≤ i ≤ α, is in F . In this work, we study S imultaneous F -D eletion for F being the family of forests. In other words, we focus on the S imultaneous F eedback V ertex S et (S im FVS) problem. Algorithmically, we show that, like its classical counterpart, S im FVS parameterized by k is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant α. In particular, we give an algorithm running in 2 O ( α k ) n O (1) time and a kernel with O (α k 3(α+1) ) vertices. The running time of our algorithm implies that S im FVS is FPT even when α ∈ o (log n ). We complement this positive result by showing that if we allow α to be in O (log n ), where n is the number of vertices in the input graph, then S im FVS becomes W[1]-hard. In particular, when α is roughly equal to c log n , for a non-zero positive constant c , the problem becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014).