A k-ranking, f, for a graph G is a function f:V(G) →{1,2,...,k} such that if u, v ∈ V(G) and f(u)=f(v), then every u − v path contains a vertex w such that f(w) ≥ f(u). In this paper, we define minimal rankings of graphs. Properties of minimal rankings are established and then used to determine χr, the minimum ranking number, and Ψr, the maximum ranking number over all minimal rankings, for complete n-partite graphs and for split graphs.
