Unique maximum matching algorithms

We consider the problem of testing the uniqueness of maximum matchings, both in the unweighted and in the weighted case. For the unweighted case, we have two results. Given a graph with n vertices and m edges, we can test whether the graph has a unique perfect matching, and find it if it exists, in O(m log⁴ n) time. This algorithm uses a recent dynamic connectivity algorithm and an old result of Kotzig characterizing unique perfect matchings in terms of bridges. Also, given one perfect matching, we can test for the existence of another in linear time. This algorithm is a modification of Edmonds' blossom-shrinking algorithm implemented using depth-first search. We prove a generalization of Kotzig's theorem characterizing unique f-factors in terms of bridges. This theorem allows us to give a modification of the first algorithm that tests whether a given graph has a unique f-factor, and find it if it exists. We also show how to modify the second algorithm to check whether a given f-factor is unique. Both extensions have the same time bounds as their perfect matching counterparts. For the weighted case, we can test in linear time whether a maximum-weight matching is unique, given the output from Edmonds' algorithm for computing such a matching. The method is an extension of our algorithm for the unweighted case.