Cutting Barnette graphs perfectly is hard

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.