  A perfect matching cut is a perfect matching that is also a cutset, or
equivalently a perfect matching containing an even number of edges on every
cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to
be NP-complete in subcubic bipartite graphs [Le & Telle, TCS '22] but its
complexity was open in planar graphs and in cubic graphs. We settle both
questions at once by showing that Perfect Matching Cut is NP-complete in
3-connected cubic bipartite planar graphs or Barnette graphs. Prior to our
work, among problems whose input is solely an undirected graph, only Distance-2
4-Coloring was known NP-complete in Barnette graphs. Notably, Hamiltonian Cycle
would only join this private club if Barnette's conjecture were refuted.
