Nonnegative signed total Roman domination in graphs

Abstract

Let $G$ be a finite and simple graph with vertex set $V(G)$. A nonnegative signed total Roman dominating function (NNSTRDF) on a graph $G$ is a function $f:V(G)\rightarrow\{-1, 1, 2\}$ 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 open 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 NNSTRDF $f$ is $\omega(f)=\sum_{v\in V (G)}f(v)$. The nonnegative signed total Roman domination number $\gamma^{NN}_{stR}(G)$ of $G$ is the minimum weight of an NNSTRDF on $G$. In this paper we initiate the study of the nonnegative signed total Roman domination number of graphs, and we present different bounds on $\gamma^{NN}_{stR}(G)$. We determine the nonnegative signed total Roman domination number of some classes of graphs. If $n$ is the order and $m$ is the size of the graph $G$, then we show that $\gamma^{NN}_{stR}(G)\ge \frac{3}{4}(\sqrt{8n+1}+1)-n$ and $\gamma^{NN}_{stR}(G)\ge (10n-12m)/5$. In addition, if $G$ is a bipartite graph of order $n$, then we prove that $\gamma^{NN}_{stR}(G)\ge \frac{3}{2}(\sqrt{4n+1}-1)-n$.