Roman Bondage Numbers of Some Graphs

A Roman dominating function on a graph $G=(V,E)$ is a function $f: V\to {0,1,2}$satisfyingtheconditionthateveryvertex$u$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 $G$. This paper determines the exact value of the Roman bondage numbers of two classes of graphs, complete $t$-partite graphs and $(n-3)$-regular graphs with order $n$ for any $n\ge 5$.