Irreducibility of the Fermi surface for planar periodic graph operators

We prove that the Fermi surface of a connected doubly periodic self-adjoint discrete graph operator is irreducible at all but finitely many energies provided that the graph (1) can be drawn in the plane without crossing edges, (2) has positive coupling coefficients, (3) has two vertices per period. If "positive" is relaxed to "complex:' the only cases of reducible Fermi surface occur for the graph of the tetralcis square tiling, and these can be explicitly parameterized when the coupling coefficients are real. The irreducibility result applies to weighted graph Laplacians with positive weights