This paper studies a nondiscrete generalization of Γ (G), the maximum cardinality of a minimal dominating set in a graph G=(V, E). In particular, a real-valued function⨍: V→[0, 1], is dominating if for each vertex υ ϵ V, the sum of the values assigned to the vertices in the closed neighborhood of υ, N [υ], is at least one, ie,⨍(N [υ])≥ 1. The weight of a dominating function⨍ is⨍(V), the sum of all values⨍(υ) for υ ϵ V, and Γ⨍(G), is the maximum weight over all minimal dominating functions. In this paper we show that:(1) Γ⨍(G) is computable and is always a rational number;(2) the decision problems corresponding to the problems of computing Γ (G) and Γ⨍(G) are NP-complete;(3) for trees Γ⨍= Γ, which implies that the value of Γ⨍ can be computed in linear time.