We show, by a careful analysis of the polynomial time algorithm due to Galluccio and Loebl, that the number of perfect matchings in a graph of small O(log n) genus can be counted in NC. So perfect matchings in small genus bipartite graphs can also be found via this approach.

We then present a different algorithm for finding a perfect matching in a planar bipartite graph. This algorithm is substantially different from the algorithm described above, and also from the algorithm of Miller and Naor, which predates the approach of Goldberg et al. and tackles the same problem. Our new algorithm extends to small genus bipartite graphs, but not to K_{3,3}-free bipartite graphs. We next show that a non-trivial extension of this algorithm allows us to compute a vertex of the fractional perfect matching polytope (such a vertex is either a perfect matching or a half-integral matching) in NC, provided the graph is planar or small genus but not necessarily bipartite, and has a perfect matching to begin with. This extension rekindles the hope for an NC-algorithm to find a perfect matching in a non-bipartite planar graph.