Parameterized Complexity of Fair Feedback Vertex Set Problem

Abstract

Given a graph \(G=(V,E)\), a subset \(S\subseteq V(G)\) is said to be a feedback vertex set of G if \(G-S\) is a forest. In the Feedback 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. This problem is extremely well studied in the realm of parameterized complexity. In this paper, we study three variants of the FVS problem: Unrestricted Fair FVS, Restricted Fair FVS, and Relax Fair FVS. In Unrestricted Fair FVS problem, we are given a graph G and a positive integer \(\ell \), the question is does there exists a feedback vertex set \(S\subseteq V(G)\) (of any size) such that for every vertex \(v\in V(G)\), v has at most \(\ell \) neighbours in S. First, we study Unrestricted Fair FVS from different parameterizations such as treewidth, treedepth and neighbourhood diversity and obtain several results (both tractability and intractability). Next, we study Restricted Fair FVS problem, where we are also given an integer k in the input and we demand the size of S to be at most k. This problem is trivially NP-complete; we show that Restricted Fair FVS problem when parameterized by the solution size k and the maximum degree \(\varDelta \) of the graph G, admits a kernel of size \(\mathcal {O}((k+\varDelta )^2)\). Finally, we study Relax Fair FVS problem, where we want that the size of S is at most k and for every vertex outside S, that is, for all \(v\in V(G)\setminus S\), v has at most \(\ell \) neighbours in S. We give an FPT algorithm for Relax Fair FVS problem running in time \(c^k n^{\mathcal {O}(1)}\), for a fixed constant c.