Dominating functions in graphs

For an arbitrary subset P of the reals, we define a function f. V → P to be a 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 v ε V, f(N (v) U (v)) ≥ 1. The P-domination number of a graph G is defined to be the infimum of f(V) taken over all Pdominating functions f. When P = (0, 1) we obtain the standard domination number. When P = [0, 1], (-1, 0, 1) or (-1, 1) we obtain the fractional, minus or signed domination numbers, respectively. [n this chapter, we survey some recent results concerning dominating functions in which negative weights are allowed.