A note on the edge Roman domination in trees

Abstract

A subset $X$ of edges of a graph $G$ is called an \textit{edge dominating set} of $G$ if every edge not in $X$ is adjacent to some edge in $X$. The edge domination number $\gamma'(G)$ of $G$ is the minimum cardinality taken over all edge dominating sets of $G$. An \textit{edge Roman dominating function} of a graph $G$ is a function $f : E(G)\rightarrow \{0,1,2 \}$ such that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e') = 2.$ The weight of an edge Roman dominating function $f$ is the value $w(f)=\sum_{e\in E(G)}f(e)$. The edge Roman domination number of $G$, denoted by $\gamma_R'(G)$, is the minimum weight of an edge Roman dominating function of $G$. In this paper, we characterize trees with edge Roman domination number twice the edge domination number.