Ideal Binary Clutters, Connectivity, and a Conjecture of Seymour

A binary clutter is the family of odd circuits of a binary matroid, that is, the family of circuits that intersect with odd cardinality a fixed given subset of elements. Let A denote the 0,1 matrix whose rows are the characteristic vectors of the odd circuits. A binary clutter is ideal if the polyhedron {x ≥ 0 : Ax ≥1} is integral. Examples of ideal binary clutters are st-paths, st-cuts, T-joins or T-cuts in graphs, and odd circuits in weakly bipartite graphs. In 1977, Seymour conjectured that a binary clutter is ideal if and only if it does not containL_F7 ,O_K5, or b(O_K5 ) as a minor. In this paper, we show that a binary clutter is ideal if it does not contain five specified minors, namely the three above minors plus two others. This generalizes Guenin’s characterization of weakly bipartite graphs, as well as the theorem of Edmonds and Johnson on T-joins and T-cuts