On the Signed (Total) $k$-Domination Number of a Graph

Let k be a positive integer and G = (V,E) be a graph of minimum degree at least k − 1. A function f : V → {−1, 1} is called a signed k-dominating function of G if ∑ u∈NG[v] f(u) ≥ k for all v ∈ V . The signed k-domination number of G is the minimum value of ∑ v∈V f(v) taken over all signed k-dominating functions of G. The signed total k-dominating function and signed total k-domination number of G can be similarly defined by changing the closed neighborhood NG[v] to the open neighborhoodNG(v) in the definition. The upper signed k-domination number is the maximum value of ∑ v∈V f(v) taken over all minimal signed k-dominating functions of G. In this paper, we study these graph parameters from both algorithmic complexity and graph-theoretic perspectives. We prove that for every fixed k ≥ 1, the problems of computing these three parameters are all NP-hard. We also present sharp lower bounds on the signed kdomination number and signed total k-domination number for general graphs in terms of their minimum and maximum degrees, generalizing several known results about signed domination.