Schmidt, Jens M.Jens M.Schmidt2020-10-232020-10-232012-12-01International Colloquium on Automata, Languages, and Programming (ICALP 2012)http://hdl.handle.net/11420/7653One of the most noted construction methods of 3-vertex-connected graphs is due to Tutte and based on the following fact: Every 3-vertex-connected graph G 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 K 4 such that every intermediate graph is 3-vertex-connected. A theorem of Barnette and Grünbaum yields a similar sequence using removals of 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|). Based on this result, we give a 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 the last years. The 3-connectivity test is conceptually different from well-known linear-time tests of 3-connectivity; it uses a certificate that is easy to verify in time O(|E|). We also provide an optimal certifying test of 3-edge-connectivity.enInformatikConference Paper10.1007/978-3-642-31594-7_66Other