Independent Roman bondage of graphs

An independent Roman dominating function (IRD-function) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the conditions that (i) every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$, and (ii) the set of all vertices assigned non-zero values under $f$ is independent. The weight of an IRD-function is the sum of its function values over all vertices, and the independent Roman domination number $i_{R}(G)$ of $G$ is the minimum weight of an IRD-function on $G$. In this paper, we initiate the study of the independent Roman bondage number $b_{iR}(G)$ of a graph $G$ having at least one component of order at least three, defined as the smallest size of set of edges $F\subseteq E(G)$ for which $i_{R}(G-F)>i_{R}(G)$. We begin by showing that the decision problem associated with the independent Roman bondage problem is NP-hard for bipartite graphs. Then various upper bounds on $b_{iR}(G)$ are established as well as exact values on it for some special graphs. In particular, for trees $T$ of order at least three, it is shown that $b_{iR}(T)\leq3,$ while for connected planar graphs the upper bounds are in terms of the maximum degree with refinements depending on the girth of the graph.