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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
##### Keywords:
binary matroid; signed matroid; binary clutter; ideal clutter; minor
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