Dominating functions with integer values in graphs—a survey

For an arbitrary subset \(\mathcal{P}\) of the reals, a function f: V → \(\mathcal{P}\) is defined to be a \(\mathcal{P}\)-dominating function of a graph G = (V,E) if the sum of its function values over any closed neighbourhood is at least 1. That is, for every υ ∊ V, f(N[υ]) ≥ 1. The definition of total \(\mathcal{P}\)-dominating function is obtained by simply changing ‘closed’ neighborhood N[υ] in the definition of \(\mathcal{P}\)-dominating function to ‘open’ neighborhood N(υ). The (total) \(\mathcal{P}\)-domination number of a graph G is defined to be the infimum of weight w(f) = \(\sum\limits_{v \in V} {f(v)} \) taken over all (total) \(\mathcal{P}\)-dominating function f. Similarly, the \(\mathcal{P}\)-edge and \(\mathcal{P}\)-star dominating functions can be defined. In this paper we survey some recent progress on the topic of dominating functions in graph theory. Especially, we are interested in \(\mathcal{P}\)-, \(\mathcal{P}\)-edge and \(\mathcal{P}\)-star dominating functions of graphs with integer values.