Cutting Barnette graphs perfectly is hard

TL;DR

This paper proves that the problem of finding a perfect matching cut is NP-complete in certain highly restricted classes of graphs, including planar, cubic, bipartite, and Barnette graphs, resolving open complexity questions.

Contribution

It establishes NP-completeness of Perfect Matching Cut in 3-connected cubic bipartite planar graphs and Barnette graphs, advancing understanding of its computational difficulty.

Findings

NP-complete in 3-connected cubic bipartite planar graphs

NP-complete in Barnette graphs

Extends complexity results to new graph classes

Abstract

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.