Ideal clutters that do not pack

For a clutter C over ground set E, a pair of distinct edges e, f ∈ E are coexclusive if every minimal cover contains at most one of them. An identification of C is another clutter obtained after identifying coexclusive edges of C. If a clutter is non-packing, then so is any identification of it. Inspired by this observation, and impelled by the lack of a qualitative characterization for ideal minimally non-packing (mnp) clutters, we reduce ideal mnp clutters even further by taking their identifications. In doing so, we reveal chains of ideal mnp clutters, demonstrate the centrality of mnp clutters with covering number two, as well as provide a qualitative characterization of irreducible ideal mnp clutters with covering number two. At the core of this characterization lies a class of objects, called marginal cuboids, that naturally give rise to ideal non-packing clutters with covering number two. We present an explicit class of marginal cuboids, and show that the corresponding clutters have one of Q6, Q2,1, Q10 as a minor, where Q6, Q2,1 are known ideal mnp clutters, and Q10 is a new ideal mnp clutter.