Quasi-total Roman domination in graphs

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 $ω (f) = f (V) = \sum_{u \in V} f (u)$ . 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.

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 $ω (f) = f (V) = \sum_{u \in V} f (u)$ . 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.

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