Mixed Double Roman Domination Number of a Graph

Abstract:

Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. A{\em mixed double Roman dominating function} $(MDRDF)$ of $G$ is a function $f:V\cup E\rightarrow \{0,1,2,3\}$ satisfying the condition every element $x\in V\cup E$ for which $f(x)=0$, is adjacent or incident to at least two elements $y,y\in V\cup E$ for which $f(y)=f(y)=2$ or one element $y"\in V\cup E$ with $f(y")=3$, and if $f(x)=1$, then element $x\in V\cup E$ must have at least one neighbor $y\in V\cup E$ with $f(y)\ge 2$. The mixed double Roman dominating number of $G$, denoted by $\gamma^*_{dR}(G)$. The weight of a $MDRDF$ $f$ is $w(f)=\sum\limits_{x\in V\cup E}f(x)$. The mixed double Roman domination number of $G$ is the minimum weight of a mixed double Roman dominating function of $G$.

Keywords:

Double Roman dominating function, Double Roman domination number, Mixed double Roman dominating function, Mixed double Roman domination number.