On the lower bounds of partial signed domination number of graphs

Let G = (V, E) be a simple graph. For any real valued function f : V -> R and S subset of V, let f(S) = Sigma(v is an element of S) f(v). Let c, d be positive integers such that gcd(c, d) = 1 and 0 < c/d <= 1 . A c/d-dominating function(partial signed dominating function) is a function f : V -> {- 1, 1} such that f (N[v]) >= 1 for at least c/d of the vertices v is an element of V. The c/d-domination number(partial signed domination number) of G is gamma(c/d) (G) = min {f (V)vertical bar f is a c/d-dominating function on G}. In this paper, we obtain a few lower bounds of gamma(c/d) (G).