  We prove a complexity dichotomy theorem for a class of Holant problems on
planar 3-regular bipartite graphs. The complexity dichotomy states that for
every weighted constraint function $f$ defining the problem (the weights can
even be negative), the problem is either computable in polynomial time if $f$
satisfies a tractability criterion, or \#P-hard otherwise. One particular
problem in this problem space is a long-standing open problem of Moore and
Robson on counting Cubic Planar X3C. The dichotomy resolves this problem by
showing that it is \numP-hard. Our proof relies on the machinery of signature
theory developed in the study of Holant problems. An essential ingredient in
our proof of the main dichotomy theorem is a pure graph-theoretic result:
Excepting some trivial cases, every 3-regular plane graph has a planar 3-way
edge perfect matching. The proof technique of this graph-theoretic result is a
combination of algebraic and combinatorial methods.
  The P-time tractability criterion of the dichotomy is explicit. Other than
the known classes of tractable constraint functions (degenerate, affine,
product type, matchgates-transformable) we also identify a new infinite set of
P-time computable planar Holant problems; however, its tractability is not by a
direct holographic transformation to matchgates, but by a combination of this
method and a global argument. The complexity dichotomy states that everything
else in this Holant class is \#P-hard.
