Total Roman domination in the lexicographic product of graphs

Abstract

A total Roman dominating function of a graph $G=(V,E)$ is a function $f: V(G)\to \{0,1,2\}$ such that for every vertex $v$ with $f(v)=0$ there exists a vertex $u$ adjacent to $v$ with $f(u)=2$, and such that the subgraph induced by the set of vertices labeled one or two has no isolated vertices. The total Roman domination number of $G$ is the minimum value of the sums $\sum_{v\in V}{f(v)}$ over all total Roman dominating functions $f$ of $G$. In this paper we study the total Roman domination number of the lexicographic product of graphs.