Parameterized complexity of fair feedback vertex set problem

Given a graph G =(V, E), a subset S subset of V(G) is said to be a feedback vertex set of Gif G - Sis a forest. In theFeedback Vertex Set (FVS) problem, we are given an undirected graph G, and a positive integer k, the question is whether there exists a feedback vertex set of size at most k. In this paper, we study three variants of the FVS problem: Unrestricted Fair FVS, Restricted Fair FVS, andRelaxed Fair FVS. InUnrestricted Fair FVS, we are given a graph Gand a positive integer l, the question is does there exist a feedback vertex set S subset of V(G) (of any size) such that for every vertex v is an element of V(G), vhas at most l neighbours in S. First, we studyUnrestricted Fair FVSfrom different parameterizations such as treewidth, treedepth, and neighbourhood diversity and obtain several results (both tractability and intractability). Next, we studyRestricted Fair FVS, where we are also given an integer kin the input and we demand the size of Sto be at most k. This problem is trivially NP-complete; we show thatRestricted Fair FVSwhen parameterized by the solution size kand the maximum degree Delta of the graph G, admits a kernel of size O(Delta k). Finally, we study theRelaxed Fair FVSproblem, where we want that the size of Sis at most kand for every vertex voutside S, vhas at most l neighbours in S. We give an FPT algorithm forRelaxed Fair FVSproblem running in time c(k)n(O(1)), for a fixed constant c. (c) 2021 Elsevier B.V. All rights reserved.