Quasi-total Roman domination in graphs

Abstract

A quasi-total Roman dominating function on a graph $G=(V,E)$ is a function $f:V\to \{0,1,2\}$ satisfying the following: - every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$, and - if $x$ is an isolated vertex in the subgraph induced by the set of vertices labeled with 1 and 2, then $f(x)=1$. The weight of a quasi-total Roman dominating function is the value $\omega (f)=f(V)={\sum }_{u\in V}f(u)$. The minimum weight of a quasi-total Roman dominating function on a graph $G$ is called the quasi-total Roman domination number of $G$. We introduce the quasi-total Roman domination number of graphs in this article, and begin the study of its combinatorial and computational properties.

15 pages