To compare the complexity of the perfect matching problem for general graphs with that for planar graphs, one might try to come up with a reduction from the perfect matching problem to the planar perfect matching problem.
The obvious way to construct such a reduction is via a {\em planarizing gadget\/}, a planar graph which replaces all edge crossings of a given graph.
We show that no such gadget exists. This provides a further indication that the complexity of the two problems is different.