FPT Algorithms for FVS Parameterized by Split and Cluster Vertex Deletion Sets and Other Parameters

Abstract

A feedback vertex set in an undirected graph is a subset of vertices whose deletion results in an acyclic graph. The problem (which we call FVS) of finding a minimum (or k sized) feedback vertex set is NP-hard in general graphs, while it is polynomial time solvable in some classes of graphs including split graphs and cluster graphs. The current best fixed-parameter tractable (FPT) algorithm for determining whether a given undirected graph has a feedback vertex set of size at most k has a runtime of \({\mathcal O}^*(3.618^k)\)(\({\mathcal O}^*\) notation hides polynomial factors). We consider the parameterized complexity of feedback vertex set parameterized by (vertex deletion) distance to some polynomially solvable classes of graphs including cluster and split graphs. We call a graph G a (c, i)-graph if its vertex set can be partitioned into c cliques and i independent sets. When \(c=0\) and \(i=2\), such a graph is simply a bipartite graph where FVS is NP-hard. It can be deduced easily that FVS is NP-hard even for constant c when \(i \ge 2\). When \(c \le 1\) and \(i \le 1\), then the graph is a split graph where FVS is solvable in polynomial time. Given a graph, let k be the size of the modulator whose deletion results in a (c, i)-graph. We address the parameterized complexity of FVS parameterized by k when \(i \le 1\). Specifically we show that 1. FVS admits an FPT algorithm that runs in \({\mathcal O}^*(3.148^k)\) time, when \(c \le 1\) and \(i \le 1\) (i.e. when the modulator is a deletion set to a split graph). When \(c \ge 2\), we generalize the algorithm to one with runtime \({\mathcal O}(3.148^{k+c}\cdot n^{{\mathcal O}(c)})\). We also show that FVS is W[1]-hard when parameterized by c (i.e. the c in the exponent of n is unavoidable) if \(i \le 1\) extending a known hardness reduction for the case when \(i=0\). 2. For the special case when \(i=0\) and \(c \ge 1\), and when there are no edges across vertices in different parts (i.e. the modulator is a deletion set to a cluster graph), we give an \({\mathcal O}^*(5^k)\) algorithm.