Independent k-rainbow bondage number of graphs

For an integer k >= 1, an independent k-rainbow dominating function (IkRDF for short) on a graph G is the following conditions: (i) if g(v) = 0, then U a function g that assigns to each vertex a set of colors chosen from the subsets of {1, 2, ..., k} satisfying u is an element of N(v) g(u) = {1, . .. , k}, and (ii) the set S = {v | g(v) = 0} is an independent set. The weight of an IkRDF g is the value w(g) = Ev is an element of V(G) |f (v)|. The independent k-rainbow domination number irk(G) is the minimum weight of an IkRDF on G. In this paper, we initiate a study of the independent k-rainbow bondage number birk(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 subset of E(G) for which irk(G - F) > irk(G). We begin by showing that the decision problem associated with the independent k-rainbow bondage problem is NP-hard for general graphs fork > 2. Then various upper bounds on bir2(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 bir2(T) <= 2.