Characterization of trees with Roman bondage number 1

Abstract

Let $G = (V, E)$ be a simple undirected graph. A Roman dominating function on $G$ is a function $f: V\to \{0, 1, 2\}$ satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to at least one vertex $v$ with $f(v) = 2$. The weight of a Roman dominating function is the value $f(G) = \sum_{u\in V} f(u)$. The Roman domination number of $G$ is the minimum weight of a Roman dominating function on $G$. The Roman bondage number of a nonempty graph $G$ is the minimum number of edges whose removal results in a graph with the Roman domination number larger than that of $G$. Rad and Volkmann [9] proposed a problem that is to determine the trees $T$ with Roman bondage number $1$. In this paper, we characterize trees with Roman bondage number $1$.