Perfect Matching in General vs. Cubic Graphs:A Note on the Planar and Bipartite Cases - Bampis, E.; Giannakos, A.; Karzanov, A.; Manoussakis, Y.; Milis, I. (RAIRO. Informatique théorique et applications, 2000)
It is known that finding a perfect matching in a general graph is AC0-equivalent to finding a perfect matching in a 3-regular (i.e. cubic) graph. In this paper we extend this result to both, planar and bipartite cases. In particular we prove that the construction problem for perfect matchings in planar graphs is as difficult as in the case of planar cubic graphs like it is known to be the case for the famous Map Four-Coloring problem. Moreover we prove that the existence and construction problems for perfect matchings in bipartite graphs are as difficult as the existence and construction problems for a weighted perfect matching of O(m) weight in a cubic bipartite graph.