Certifying 3-Edge-Connectivity

TL;DR

This paper introduces a linear-time certifying algorithm for testing 3-edge-connectivity in graphs, providing constructive sequences or cuts, and extends to 3-vertex-connectivity analysis.

Contribution

It presents the first linear-time certifying algorithm for 3-edge-connectivity and methods for computing related components and representations.

Findings

Algorithm certifies 3-edge-connectivity or finds 2-edge-cuts.

Computes 3-edge-connected components in linear time.

Provides construction sequences for 3-edge-connected graphs.

Abstract

We present a certifying algorithm that tests graphs for 3-edge-connectivity; the algorithm works in linear time. If the input graph is not 3-edge-connected, the algorithm returns a 2-edge-cut. If it is 3-edge-connected, it returns a construction sequence that constructs the input graph from the graph with two vertices and three parallel edges using only operations that (obviously) preserve 3-edge-connectivity. Additionally, we show how compute and certify the 3-edge-connected components and a cactus representation of the 2-cuts in linear time. For 3-vertex-connectivity, we show how to compute the 3-vertex-connected components of a 2-connected graph.