  A Roman dominating function on a graph $G=(V,E)$ is a function
$f:V\rightarrow\{0,1,2\}$ such that every vertex $v\in V$ with $f(v)=0$ has at
least one neighbor $u\in V$ with $f(u)=2$. The weight of a Roman dominating
function is the value $f(V(G))=\sum_{u\in V(G)}f(u)$. The minimum weight of a
Roman dominating function on a graph $G$ is called the Roman domination number,
denoted by $\gamma_{R}(G)$. The Roman bondage number $b_{R}(G)$ of a graph $G$
with maximum degree at least two is the minimum cardinality of all sets
$E'\subseteq E(G)$ for which $\gamma_{R}(G-E')>\gamma_R(G)$. In this paper, we
first show that the decision problem for determining $b_{\rm R}(G)$ is NP-hard
even for bipartite graphs and then we establish some sharp bounds for $b_{\rm
R}(G)$ and characterizes all graphs attaining some of these bounds.
