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.