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The two-point Fano and ideal binary clutters. (English) Zbl 1449.05041
In this paper, necessary conditions for a binary clutter to be non-ideal are obtained. By that, the authors prove the weakened version of the flowing conjecture. Moreover, it is shown that for each element \(e\) of the ground set at least one member of a blocking pair of minimally non-ideal binary clutters has a two-point Fano minor going through \(e\), in the case when the both members of the blocking pair don’t comprise sets with cardinality 3. The proof is based on the concept of a strict \(e\)-hub.
MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
05C83 Graph minors
90C10 Integer programming
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
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