Partial signed domination in graphs

Let G = (V, E) be a graph. For any real valued function f : V → R and S ⊆ V, let f(S) = Σ_u∈sf(u). Let c, d be positive integers such that gcd(c, d) = 1 and 0 < c/d ≤ 1. A c/d-dominating function f is a function f : V → {-1, 1} such that f[v] ≥ 1 for at least c/d of the vertices V. The c/d-domination number of G, denoted by γ_c/d (G), is defined as min{f(V) | f is a c/d-dominating function on G}. We determine a sharp lower bound on γ_c/d(G) for regular graphs G, determine the value of γ_c/d for an arbitrary cycle C_n and show that the decision problem PARTIAL SIGNED DOMINATING FUNCTION is N P-complete.