The Minimally Non-Ideal Binary Clutters with a Triangle

On ideal minimally non-packing clutters

We consider the following conjecture proposed by Cornuéjols, Guenin and Margot: every ideal minimally non-packing clutter has a transversal of size 2. For a clutter C, let C̃ denote the set of hyperedges of C which intersect any minimum transversal in exactly one element. We divide the (non-)existence problem of an ideal minimally non-packing clutter D into two steps. In the first step, we give ...

Ideal clutters

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 fx 0 : Ax 1g is integral. Examples of ideal binary clutters are st-paths, st-cuts, T -joins ...

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 fx 0 : Ax 1g is integral. Examples of ideal binary clutters are st-paths, st-cuts, T -joins ...

The Two-Point Fano and Ideal Binary Clutters

Let F be a binary clutter. We prove that if F is non-ideal, then either F or its blocker b(F) has one of L7,O5,LC7 as a minor. L7 is the non-ideal clutter of the lines of the Fano plane, O5 is the non-ideal clutter of odd circuits of the complete graph K5, and the two-point Fano LC7 is the ideal clutter whose sets are the lines, and their complements, of the Fano plane that contain exactly one ...