Construction sequences of 3-connected graphs

Jens M. Schmidt

Abstract

Tutte proved 1961 that every 3-connected graph on more than 4 nodes has a contractible edge. Barnette and Grünbaum proved 1969 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 the K_4 can be computed in O(|V|^2) time by extending Barnette and Grünbaum's theorem. As an application, we derive a certificate for the 3-connectedness of graphs that can be easily computed and verified.