Nonnegative signed Roman domination in graphs

Abstract

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.