Minimal rankings

A k-ranking, f, for a graph G is a function f : V(G) --> {1, 2,..., k} such that if u, v is an element of 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 chi(r), the minimum ranking number, and psi(r), the maximum ranking number over all minimal rankings, for complete n-partite graphs and for split graphs. (C) 1995 John Wiley & Sons, Inc.