Complexity results for $k$-domination and $\alpha$-domination problems and their variants

Let G = (V,E) be a simple and undirected graph. For some integer k > 1, a set D ⊆ V is said to be a k-dominating set in G if every vertex v of G outside D has at least k neighbors in D. Furthermore, for some real number α with 0 < α 6 1, a set D ⊆ V is called an α-dominating set in G if every vertex v of G outside D has at least α×dv neighbors in D, where dv is the degree of v in G. The cardinality of a minimum k-dominating set and a minimum α-dominating set in G is said to be the k-domination number and the α-domination number of G, respectively. In this paper, we present some approximability and inapproximability results on the problem of finding of k-domination number and α-domination number of some classes of graphs. Moreover, we introduce a generalization of α-dominating set which we call f-dominating set. Given a function f : N → R, where N = {1, 2, 3, . . .}, a set D ⊆ V is said to be an f -dominating set in G if every vertex v of G outside D has at least f(dv) neighbors in D. We prove NP-hardness of the problem of finding of a minimum f -dominating set in G, for a large family of functions f .