Computing the $4$-Edge-Connected Components of a Graph in Linear Time

TL;DR

This paper introduces the first linear-time algorithm for computing 4-edge-connected components and testing 4-edge connectivity in undirected graphs, advancing graph connectivity analysis.

Contribution

It presents the first linear-time algorithms for identifying 4-edge-connected components and testing 4-edge connectivity in graphs, building on previous linear-time methods for 3-edge cuts.

Findings

First linear-time algorithm for 4-edge-connected components

Efficient testing of 4-edge connectivity

Extension of linear-time algorithms from 3-edge to 4-edge connectivity

Abstract

We present the first linear-time algorithm that computes the $4$-edge-connected components of an undirected graph. Hence, we also obtain the first linear-time algorithm for testing $4$-edge connectivity. Our results are based on a linear-time algorithm that computes the $3$-edge cuts of a $3$-edge-connected graph $G$, and a linear-time procedure that, given the collection of all $3$-edge cuts, partitions the vertices of $G$ into the $4$-edge-connected components.