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Tutte proved that every 3-connected graph on more than 4 nodes has acontractible edge. Barnette and Gruenbaum proved the existence of a removableedge in the same setting. We show that the sequence of contractions and thesequence of removals from G to the K_4 can be computed in O(|V|^2) time byextending Barnette and Gruenbaum's theorem. As an application, we derive acertificate for the 3-connectedness of graphs that can be easily computed andverified.

Construction Sequences and Certifying 3-Connectedness