On the Parameterized Complexity of Minus Domination

Dominating Set is a well-studied combinatorial problem. Given a graph \(G=(V,E)\) , a dominating function \(f:V(G)\rightarrow \{0, 1\}\) is a labeling of the vertices of G such that \(\sum _{w \in N[v]} f(w) \ge 1\) for each vertex \(v\in V(G)\) , where \(N[v]=\{v\} \cup \{u \mid uv \in E(G)\}\) . We study a generalization of Dominating Set called Minus Domination (in short, MD) where \(f: V(G) \rightarrow \{-1, 0, 1\}\) . Such a function is said to be a minus dominating function if for each vertex \(v\in V(G)\) , we have \(\sum _{w \in N[v]}f(w) \ge 1\) . The objective is to minimize the weight of a minus domination function, which is \(f(V)= \sum _{u \in V(G)}f(u)\) . The problem is NP-hard even on bipartite, planar, and chordal graphs. In this paper, we study MD from the perspective of parameterized complexity. After observing the complexity of the problem with the natural parameters such as the number of vertices labeled 1, \(-1\) and 0, we study the problem with respect to structural parameters. We show that MD is fixed-parameter tractable when parameterized by twin-cover number, neighborhood diversity or the combined parameters component vertex deletion set and size of the largest component. In addition, we give an XP-algorithm when parameterized by distance to cluster number.