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.