On the Parameterized Complexity of Minus Domination

Sriram Bhyravarapu, Lawqueen Kanesh, A. Mohanapriya, Nidhi Purohit, N. Sadagopan, Saket Saurabh

Abstract: 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.