Contractions, Removals, and Certifying 3-Connectivity in Linear Time

Abstract

One of the most noted construction methods of $3$-vertex-connected graphs is due to Tutte and is based on the following fact: Any $3$-vertex-connected graph $G=(V,E)$ on more than $4$ vertices contains a contractible edge, i.e., an edge whose contraction generates a $3$-connected graph. This implies the existence of a sequence of edge contractions from $G$ to the complete graph $K_4$, such that every intermediate graph is $3$-vertex-connected. A theorem of Barnette and Grünbaum gives a similar sequence using removals on edges instead of contractions. We show how to compute both sequences in optimal time, improving the previously best known running times of $O(|V|^2)$ to $O(|E|)$. This result has a number of consequences; an important one is a new linear-time test of $3$-connectivity that is certifying; finding such an algorithm has been a major open problem in the design of certifying algorithms in recent years. The test is conceptually different from well-known linear-time $3$-connectivity tests and uses a certificate that is easy to verify in time $O(|E|)$. We show how to extend the results to an optimal certifying test of $3$-edge-connectivity.