Edge Roman Domination in Graphs

Abstract

An $EdgeRomandominatingfunction$ of a graph $G=(V,E)$ is a function $f^{\prime }:E\to \{0,1,2\}$ satisfying the condition that every edge $x$ for which $f^{\prime }(x)=0$ is adjacent to at least one edge $y$ for which $f^{\prime }(y)=2$. The $weight$ of an Edge Roman dominating function is the value $f^{\prime }(E)=\sum _{x\in E}{f}^{\prime }(x)$. The minimum weight of an Edge Roman dominating function on a graph $G$ is called the $EdgeRomandominationnumber$ of $G$. In this paper, we initiate a study of this parameter.