On The Double Roman bondage numbers of Graphs

For a graph $G=(V,E)$, a double roman dominating function (DRDF) is a function $f : V \longrightarrow \{0, 1, 2,3\}$ having the property that if $f(v)=0$ for some vertex $v$, then $v$ has at least two neighbors assigned $2$ under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$ then $v$ has at least one neighbor $w$ with $f(w) \geq 2$. The weight of a DRDF $f$ is the sum $f (V) =\sum_{u\in V} f (u)$. The minimum weight of a DRDF on a graph $G$ is the double Roman domination number of $G$ and is denoted by $\gamma_{dR}(G)$. The double roman bondage number of $G$, denoted by $b_{dR}(G)$, is the minimum cardinality among all edge subsets $B \subseteq E(G)$ such that $\gamma_{dR}(G-B) > \gamma_{dR}(G)$. In this paper we study the double roman bondage number in graphs. We determine the double roman bondage number in several families of graphs, and present several bounds for the double roman bondage number. We also study the complexity issue of the double roman bondage number and prove that the decision problem for the double roman bondage number is NP-hard even when restricted to bipartite graphs.