Conflict Free Version of Covering Problems on Graphs: Classical and Parameterized

Let \(\mathrm{\Pi }\) be a family of graphs. In the classical \(\mathrm{\Pi }\)-Vertex Deletion problem, given a graph G and a positive integer k, the objective is to check whether there exists a subset S of at most k vertices such that \(G-S\) is in \(\mathrm{\Pi }\). In this paper, we introduce the conflict free version of this classical problem, namely Conflict Free \(\mathrm{\Pi }\)-Vertex Deletion (CF-\(\mathrm{\Pi }\)-VD), and study these problems from the viewpoint of classical and parameterized complexity. In the CF-\(\mathrm{\Pi }\)-VD problem, given two graphs G and H on the same vertex set and a positive integer k, the objective is to determine whether there exists a set \(S\subseteq V(G)\), of size at most k, such that \(G-S\) is in \(\mathrm{\Pi }\) and H[S] is edgeless. Initiating a systematic study of these problems is one of the main conceptual contribution of this work. We obtain several results on the conflict free version of several classical problems. Our first result shows that if \(\mathrm{\Pi }\) is characterized by a finite family of forbidden induced subgraphs then CF-\(\mathrm{\Pi }\)-VD is Fixed Parameter Tractable (FPT). Furthermore, we obtain improved algorithms for conflict free version of several well studied problems. Next, we show that if \(\mathrm{\Pi }\) is characterized by a “well-behaved” infinite family of forbidden induced subgraphs, then CF-\(\mathrm{\Pi }\)-VD is W[1]-hard. Motivated by this hardness result, we consider the parameterized complexity of CF-\(\mathrm{\Pi }\)-VD when H is restricted to well studied families of graphs. In particular, we show that the conflict free versions of several well-known problems such as Feedback Vertex Set, Odd Cycle Transversal, Chordal Vertex Deletion and Interval Vertex Deletion are FPT when H belongs to the families of d-degenerate graphs and nowhere dense graphs.