Further results on the signed Italian domination

Abstract

A signed Italian dominating function on a graph $$G=(V,E)$$ is a function $$f: V\rightarrow \{-1, 1, 2\}$$ satisfying the condition that for every vertex u, $$f[u]\ge 1$$ . The weight of the signed Italian dominating function, f, is the value $$f(V)=\sum _{u\in V}f(u)$$ . The signed Italian dominating number of a graph G, denoted by $$\gamma _{sI}(G)$$ , is the minimum weight of a signed Italian dominating function on a graph G. In this paper, we prove that for any tree T of order $$n\ge 2$$ , $$\gamma _{sI}(T)\ge \frac{-n+4}{2}$$ and we characterize all trees attaining this bound. In addition, we obtain some results about the signed Italian domination number of some graph operations. Furthermore, we prove that the signed Italian domination problem is $$\mathbf {NP}$$ -Complete for bipartite graphs.