Computing Vertex-Disjoint Paths using MAOs

TL;DR

This paper introduces a linear-time algorithm for computing vertex-disjoint paths between specific vertex pairs in a graph, improving upon previous flow-based methods and simplifying existing proofs.

Contribution

It presents a novel linear-time algorithm for finding vertex-disjoint paths between pairs in graphs using MAOs, enhancing efficiency and theoretical understanding.

Findings

Algorithm runs in linear time O(n+m)

Improves upon previous flow-based solutions

Simplifies proofs related to pendant pairs

Abstract

Let G be a graph with minimum degree $\delta$. It is well-known that maximal adjacency orderings (MAOs) compute a vertex set S such that every pair of S is connected by at least $\delta$ internally vertex-disjoint paths in G. We present an algorithm that, given any pair of S, computes these $\delta$ paths in linear time O(n+m). This improves the previously best solutions for these special vertex pairs, which were flow-based. Our algorithm simplifies a proof about pendant pairs of Mader and makes a purely existential proof of Nagamochi algorithmic.