Fractional Dominating Parameters

For an arbitrary subset P of the reals and a graph G with vertex set V, a function f: V→ P is a P-dominating function of G if the sum of the function values over any closed neighborhood is at least 1. That is, for every v ∈ V, f(N[v]) ≥ 1. The P-domination number of a graph G is defined to be the infimum of f(V ) taken over all P-dominating functions f. When P= { 0, 1 } we obtain the standard domination number, and when P= { − 1, 0, 1 } or {−1, 1} we obtain the minus or signed domination number. In this chapter, we survey some results concerning fractional dominating parameters when P= [ 0, 1 ] or ℤ or ℝ in which case we obtain the fractional, integer, or real domination numbers, respectively.