Rankings of Graphs

A vertex (edge) coloring φ : V → {1, 2, . . . , t} (φ′ : E → {1, 2, . . . , t}) of a graph G = (V,E) is a vertex (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 χr(G) (edge ranking number χ′ r (G)) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we study the algorithmic complexity of the Vertex Ranking and Edge Ranking problems. It is shown that χr(G) can be computed in polynomial time when restricted to graphs with treewidth at most k for any fixed k. We characterize the graphs where the vertex ranking number χr and the chromatic number χ coincide on all induced subgraphs, show that χr(G) = χ(G) implies χ(G) = ω(G) (largest clique size), and give a formula for χ′ r (Kn).