  In 1988, Vazirani gave an NC algorithm for computing the number of perfect
matchings in $K_{3,3}$-minor-free graphs by building on Kasteleyn's scheme for
planar graphs, and stated that this "opens up the possibility of obtaining an
NC algorithm for finding a perfect matching in $K_{3,3}$-free graphs." In this
paper, we finally settle this 30-year-old open problem. Building on recent NC
algorithms for planar and bounded-genus perfect matching by Anari and Vazirani
and later by Sankowski, we obtain NC algorithms for perfect matching in any
minor-closed graph family that forbids a one-crossing graph. This family
includes several well-studied graph families including the $K_{3,3}$-minor-free
graphs and $K_5$-minor-free graphs. Graphs in these families not only have
unbounded genus, but can have genus as high as $O(n)$. Our method applies as
well to several other problems related to perfect matching. In particular, we
obtain NC algorithms for the following problems in any family of graphs (or
networks) with a one-crossing forbidden minor:
  $\bullet$ Determining whether a given graph has a perfect matching and if so,
finding one.
  $\bullet$ Finding a minimum weight perfect matching in the graph, assuming
that the edge weights are polynomially bounded.
  $\bullet$ Finding a maximum $st$-flow in the network, with arbitrary
capacities.
  The main new idea enabling our results is the definition and use of
matching-mimicking networks, small replacement networks that behave the same,
with respect to matching problems involving a fixed set of terminals, as the
larger network they replace.
