On Whitney's 2‐isomorphism theorem for graphs

Abstract

AbstractLet G and H be 2‐connected 2‐isomorphic graphs with n nodes. Whitney's 2‐isomorphism theorem states that G may be transformed to a graph G* isomorphic to H by repeated application of a simple operation, which we will term “switching”. We present a proof of Whitney's theorem that is much shorter than the original one, using a graph decomposition by Tutte. The proof also establishes a surprisingly small upper bound, namely n‐2, on the minimal number of switchings required to derive G* from G. The bound is sharp in the sense that for any integer N there exist graphs G and H with n ≥ N nodes for which the minimal number of switchings is n‐2.