Signed total Roman domination in graphs

Abstract

Let $G$ be a finite and simple graph with vertex set $V(G)$. A signed total Roman dominating function (STRDF) on a graph $G$ is a function $f:V(G)\rightarrow \{-1,1,2\}$ satisfying the conditions that (i) $\sum _{x\in N(v)}f(x)\ge 1$ for each vertex $v\in V(G)$, where $N(v)$ is the neighborhood of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ is adjacent to at least one vertex $v$ for which $f(v)=2$. The weight of an SRTDF $f$ is $\sum _{v\in V(G)}f(v)$. The signed total Roman domination number $\gamma _{stR}(G)$ of $G$ is the minimum weight of an STRDF on $G$. In this paper we initiate the study of the signed total Roman domination number of graphs, and we present different bounds on $\gamma _{stR}(G)$. In addition, we determine the signed total Roman domination number of some classes of graphs.