  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 tetrakis 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.
