Another simple proof of the validity of Nagamochi and Ibaraki's min-cut algorithm and Queyranne's extension to symmetric submodular function minimization

M. Stoer and F. Wagner and independently A. Frank have found a simple proof of the validity of Nagamochi and Ibaraki's min-cut algorithm. This note points out some nice property of the behavior of Nagamochi and Ibaraki's min-cut algorithm, which also gives another simple proof of the validity of their algorithm. The proof relies only on the symmetric submodularity of the cut function. Hence, it also gives another simple proof of the validity of Queyranne's extension of Nagamochi and Ibaraki's algorithm to symmetric submodular function minimization.