On 3-cut reductions of minimally 1-factorable cubic bigraphs Original Research Article

A cubic bigraph G is minimally 1-factorable if every 1-factor lies in precisely one 1-factorization. We characterize 3-bridges of G and prove that the 3-cut reductions of G are still minimally 1-factorable; thus, the open classification problem is reduced to the study of 3-bridge-free minimally 1-factorable cubic bigraphs. Furthermore, we prove that if ab, cd are edges of G such that the graph obtained by twisting them in ad, bc is still minimally 1-factorable, then ab, cd lie in some 3-bridge of G.