Rankings of graphs

A vertex (edge) coloring c: V!f1; 2;:::;tg (c 0: E!f1; 2;:::; tg) of a graph G =(V;E) isavertex (edge) t-ranking if for any two vertices (edges) of the same color every path between them contains a vertex (edge) of larger color. The vertex ranking number r(G) (edge ranking number 0 r(G)) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we study the algorithmic complexityofthevertex ranking and edge ranking problems. Among others it is shown that r(G) can be computed in polynomial time when restricted to graphs with treewidth at most k for any xed k. We characterize those graphs where the vertex ranking number r and the chromatic number coincide on all induced subgraphs, show that r(G) = (G) implies (G) =!(G) (largest clique size) and give a formula for 0 r(K n)