Optimal vertex ranking of block graphs

A vertex ranking of an undirected graph G is a labeling of the vertices of G with integers such that every path connecting two vertices with the same label i contains an intermediate vertex with label j>i. A vertex ranking of G is called optimal if it uses the minimum number of distinct labels among all possible vertex rankings. The problem of finding an optimal vertex ranking for general graphs is NP-hard, and NP-hard even for chordal graphs which form a superclass of block graphs. In this paper, we present the first polynomial algorithm which runs in O(n2logΔ) time for finding an optimal vertex ranking of a block graph G, where n and Δ denote the number of vertices and the maximum degree of G, respectively.