Contractions, Removals and How to Certify 3-Connectivity in Linear Time

TL;DR

This paper presents a linear-time algorithm for certifying 3-connectivity in graphs, improving previous methods and providing efficient, verifiable certificates for 3- and 3-edge connectivity.

Contribution

It introduces an optimal linear-time algorithm for computing contraction and removal sequences certifying 3-connectivity, solving an open problem and extending to 3-edge connectivity.

Findings

Linear-time certifying 3-connectivity test

Optimal algorithms for contraction and removal sequences

Efficient verifiable certificates for connectivity

Abstract

It is well-known as an existence result that every 3-connected graph G=(V,E) on more than 4 vertices admits a sequence of contractions and a sequence of removal operations to K_4 such that every intermediate graph is 3-connected. We show that both sequences can be computed in optimal time, improving the previously best known running times of O(|V|^2) to O(|V|+|E|). This settles also the open question of finding a linear time 3-connectivity test that is certifying and extends to a certifying 3-edge-connectivity test in the same time. The certificates used are easy to verify in time O(|E|).