A Fast Algorithm for Enumerating Non-Bipartite Maximal Matchings

For a graph G = (V, E), a stable set in G is a vertex set such that no pair of vertices in the set are connected by an edge. Stable set enumeration problems have been studied because of their applications to optimization, computational geometry, etc. However, the problem of speeding up enumeration algorithms for stable sets is still open. In this paper, we consider the problem of enumerating all maximal matchings of a given non-bipartite graph G = (V, E), which is a special case of the stable set enuemration problem, and propose an algorithm with a simple structure. By applying the stable set enumeration algorithms to this problem, the computation time is O(|V ||E|2N). Our algorithm runs in O(|E| + |V | +∆N) time, very fast compared with those algorithms. Here N denotes the number of maximal matchings in G, and ∆ denotes the maximum degree of G.