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.

翻译：完美匹配割是指既是完美匹配又是割集的边集，等价于在每个环上包含偶数条边的完美匹配。对应的算法问题"完美匹配割"已知在次三次二部图中是NP完全的[Le & Telle, TCS '22]，但在平面图和三次图中的复杂性此前尚未解决。我们通过证明完美匹配割在3-连通三次二部平面图（即Barnette图）中是NP完全的，一次性解决了这两个问题。在我们之前，输入仅为无向图的问题中，只有距离2的4染色问题已知在Barnette图中是NP完全的。值得注意的是，哈密顿回路问题只有推翻Barnette猜想才会加入这个专属俱乐部。