On Total Roman Domination in Graphs

Abstract

A Roman dominating function (RDF) on a graph $G = (V,E)$ is a function $f:V \rightarrow \lbrace 0,1,2\rbrace$ satisfying the condition that every vertex u for which $f(u) = 0$ is adjacent to at least one vertex v for which $f(v)=2$. A total Roman dominating function on a graph $G = (V,E)$ is a Roman dominating function $f : V \rightarrow \lbrace 0,1,2 \rbrace$ satisfying the condition that every vertex u for which $f(u) > 0$ is adjacent to at least one vertex v for which $f(v) > 0$. The weight of a total Roman dominating function is the value $\displaystyle f(V)= \sum _{u \in V} f(u)$. The minimum weight of a total Roman dominating function on a graph G is called the total Roman domination number of G and denoted by $\gamma _{tR}(G)$. In this paper, we establish some bounds on the Total Roman domination number in terms of its order and girth.

The original version of this chapter was revised: The name of the second author was corrected. An erratum to this chapter can be found at https://doi.org/10.1007/978-3-319-64419-6_58