Minus k-subdomination in graphs III

Let G = (V, E) be a graph. For any real valued function 1: V--+ R and S ~ V, let 1 (S) = L:UES 1 ( u). The weight of 1 is defined as I(V). We will also denote I(N[v]) by I[v] ' where v E V. A minus k-subdominating function (k8F) for G is defined in [1] as a function 1: V--+ {-I, 0, I} such that I[v] 2:: 1 for at least k vertices of G. The minus k-subdomination number of a graph G, denoted by 'Yks 101 (G), is equal to min{I(V) I f is a minus k8F of G}. Hattingh and Ungerer show in [5] that if T is a tree of order n 2:: 2 and k is an integer such that 1:::; k::; n- 1, then 'Yk/01(T) 2:: k- n + 2; In this paper, we characterise trees which achieve the lower bound, and show that the decision problem corresponding to the computation of this parameter is NP-complete.