Nonnegative signed Roman domination in graphs

Let \(G\) be a finite and simple graph with vertex set \(V(G)\). A nonnegative signed Roman dominating function (NNSRDF) on a graph \(G\) is a function \(f:V(G)\to \{-1,1,2\}\) satisfying the conditions that (i) \(\sum_{x\in N[v]}f(x)\ge 0\) for each \(v \in V(G)\), where \(N[v]\) is the closed neighborhood of \(v\) and (ii)every vertex u for which \(f(u)=-1\) has a neighbor v for which \(f(v)=2\). The weight of an NNSRDF \(f\) is \(\omega(f) = \sum_{v\in V(G)} f(v)\). The nonnegative signed Roman domination number \(\gamma_{sR}^{NN} (G)\) of \(G\) is the minimum weight of an NNSRDF \(G\) In this paper, we initiate the study of the nonnegative signed Roman domination number of a graph and we present different bounds on \(\gamma _{sR}^{NN}(G) \ge (8n-12m)/7\). In addition, if \(G\) is a bipartite graph of order \(n\), then we prove that \(\gamma _{sR}^{NN}(G) \ge^\frac{3}{2}(\sqrt{4n+9}-3)-n\), and we characterize the external graphs.