Graphs with large rank numbers and rank numbers of subdivided stars

Abstract

A k-ranking of a graph G is a function f : V (G) → {1, 2,. .. , k} such that if f (u) = f (v) then every uv path contains a vertex w such that f (w) > f (u). The rank number of G, denoted χ r (G), is the minimum k such that a k-ranking exists for G. It is known that given a graph G and a positive integer t the question of whether χ r (G) ≤ t is NP-complete. In this paper we characterize graphs with large rank numbers. In addition, we characterize subdivided stars based on their rank numbers.