Minimal rankings and the arank number of a path

Given a graph G, a function f: V (G) → {1, 2,..., k} is a k-ranking of G if f(u) = f(v) implies every u − v path contains a vertex w such that f(w)> f(u). A k-ranking is minimal if the reduction of any label greater than 1 violates the described ranking property. The arank number of a graph, denoted ψr(G), is the largest k such that G has a minimal k-ranking. We present new results involving minimal k-rankings of paths. In particular we determine ψr(Pn), a problem suggested by Laskar and Pillone in 2000.