Independent Italian bondage of graphs

An independent Italian dominating function (IID-function) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the conditions that (i) ∑u∈N(v) f(u) ≥ 2 when f(v) = 0, and (ii) the set of all vertices assigned non-zero values under f is independent. The weight of an IID-function is the sum of its function values over all vertices, and the independent Italian domination number iI (G) of G is the minimum weight of an IID-function on G. In this paper, we initiate the study of the independent Italian bondage number biI (G) of a graph G having at least one component of order at least three, defined as the smallest size of a set of edges of G whose removal from G increases iI (G). We show that the decision problem associated with the independent Italian bondage problem is NP-hard for arbitrary graphs. Moreover, various upper bounds on biI (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 biI (T) ≤ 2.