Quasi-total Roman Domination in Graphs

Abstract

A quasi-total Roman dominating function on a graph $G=(V, E)$ is a function $f : V \rightarrow \{0,1,2\}$ 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.