Diameter Determination on Restricted Graph Faminlies

Determining the diameter of a graph is a fundamental graph operation, yet no e/cient (i.e. linear or quadratic time) algorithm is known. In this paper, we examine the diameter problem on chordal graphs and AT-free graphs and show that a very simple (linear time) 2-sweep LexBFS algorithm identi9es a vertex of maximum eccentricity unless the given graph has a speci9ed induced subgraph (it was previously known that a single LexBFS algorithm is guaranteed to end at a vertex that is within 1 of the diameter for chordal graphs and AT-free graphs). As a consequence of the forbidden induced subgraph result on chordal graphs, our algorithm is guaranteed to work optimally for directed path graphs (it was previously known that a single LexBFS algorithm is guaranteed to work optimally for interval graphs). ? 2001 Elsevier Science B.V. All rights reserved.