Tutte proved that every 3‐vertex‐connected graph G on more than 4 vertices has a contractible edge. Barnette and Grünbaum proved the existence of a removable edge in the same setting. We show that the sequence of contractions and the sequence of removals from G to K
4 can be computed in O(|V|2) time by extending Barnette’s and Grünbaum’s theorem. As an application, we derive a certificate for the 3‐vertex‐connectivity of graphs that can be easily computed and verified.
