The Schrijver system of odd join polyhedra

Graphs for which the set oft-joins andt-cuts has “the max-flow-min-cut property”, i.e. for which the minimal defining system of thet-join polyhedron is totally dual integral, have been characterized by Seymour. An extension of this problem isto characterize the (uniquely existing) minimal totally dual integral defining system (Schrijver-system) of an arbitrary t-join polyhedron. This problem is solved in the present paper. The main idea is to uset-borders, a generalization oft-cuts, to obtain an integer minimax formula for the cardinality of a minimumt-join. (At-border is the set of edges joining different classes of a partition of the vertex set intot-odd sets.) It turns out that the (uniquely existing) “strongest minimax theorem” involves just this notion.

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