CONTRACTIONS, REMOVALS, AND CERTIFYING 3-CONNECTIVITY IN LINEAR TIME

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 Grunbaum 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(vertical bar V vertical bar(2)) to O(vertical bar E vertical bar). 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(vertical bar E vertical bar). We show how to extend the results to an optimal certifying test of 3-edge-connectivity.