Roman bondage in graphs

A Roman dominating function on a graph G is a function f:V(G) → {0,1,2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value ${\displaystyle f\left(V\left(G\right)\right)={\sum }_{u\in V\left(G\right)}f\left(u\right)}$. The Roman domination number, ${\displaystyle {\gamma }_R\left(G\right)}$, of G is the minimum weight of a Roman dominating function on G. In this paper, we define the Roman bondage ${\displaystyle b_R\left(G\right)}$ of a graph G with maximum degree at least two to be the minimum cardinality of all sets E' ⊆ E(G) for which ${\displaystyle {\gamma }_R(G-E\prime )>{\gamma }_R\left(G\right)}$. We determine the Roman bondage number in several classes of graphs and give some sharp bounds.