Ideal 0, 1 Matrices

We define a 0, 1 matrix M to be ideal if all vertices of the polyhedron { x: Mx ≥ 1, x ≥ 0 } have only 0, 1 components. We expand the list of known minor minimal nonideal matrices by several thousand. Many of these examples are obtainedpolyhedrally, by constructing new minimally nonideal matrices from old ones. We present a conjecture that might be viewed as the counterpart for ideal matrices of Berge′s strong perfect graph conjecture. We provide evidence for the conjecture bycompletely characterizing all minimally nonideal circulants.