A Characterization of Box 1d-Integral Binary Clutters

Let Q6 denote the port of the dual Fano matroid F*7 and let Q7 denote the clutter consisting of the circuits of the Fano matroid F7 that contain a given element. Let L be a binary clutter on E and let d ≥ 2 be an integer. We prove that all the vertices of the polytope {x ∈ RE+ | x(C) ≥ 1 for C ∈ L} ∩ {x | a ≤ x ≤ b} are 1d-integral, for any 1d-integral a, b, if and only if L does not have Q6 or Q7 as a minor. This includes the class of ports of regular matroids. Applications to graphs are presented, extending a result from Laurent and Pojiak [7].