Roman Bondage Number of a Graph

The Roman dominating function on a graph $G=(V,E)$ is a function $f: V\rightarrow{0,1,2}$suchthateachvertex$x$with$f(x)=0$ is adjacent to at least one vertex $y$ with $f(y)=2$. The value $f(G)=\sum\limits_{u\in V(G)} f(u)$iscalledtheweightof$f$\mathrm{.}TheRomandominationnumber$\gamma_{\rm R}(G)$ is defined as the minimum weight of all Roman dominating functions. This paper defines the Roman bondage number $b_{\mathrm{R}}(G)$ of a nonempty graph $G=(V,E)$ to be the cardinality among all sets of edges $B\subseteq E$ for which ${\gamma }_{\mathrm{R}}(G-B)>{\gamma }_{\mathrm{R}}(G)$. Some bounds are obtained for $b_{\mathrm{R}}(G)$, and the exact values are determined for several classes of graphs. Moreover, the decision problem for $b_{\mathrm{R}}(G)$ is proved to be NP-hard even for bipartite graphs.