Complexity of diameter on AT-free graphs is linear

TL;DR

This paper introduces a linear time algorithm for computing the diameter of AT-free graphs, updates the concept of polar pairs, and improves runtimes for related graph problems.

Contribution

It presents a novel linear time algorithm for diameter computation in AT-free graphs and refines the understanding of polar pairs and related graph properties.

Findings

Linear time diameter algorithm for AT-free graphs

New properties of polar pairs in dominating pair graphs

Improved runtime for finding simplicial vertices in general graphs

Abstract

We develop a linear time algorithm for finding the diameter of an asteroidal triple-free (AT-free) graph. Furthermore, we update the definition of polar pairs and develop new properties of polar pairs for (weak) dominating pair graphs. We prove that the problem of computing a simplicial vertex in a general graph can be accomplished in O(n^2) based on an existing reduction to the problem of finding diameter in an AT-free graph. We improve the best-known run-time complexities of several graph theoretical problems.