Real and integer domination in graphs

For an arbitrary subset ℘ of the reals, we define a function f : V → ℘ to be a ℘-dominating function of graph G = (V,E) if the sum of the function values over any closed neighbourhood is at least 1. That is, for every v ∈ V, f(N(v)∪{v}) ≥ 1. The ℘-domination number of a graph is defined to be the infimum of f(V) taken over all ℘-dominating functions f. When ℘ = {0,1} one obtains the standard domination number. We obtain various theoretical and computational results on the ℘-domination number of a graph.