The Minimally Non-Ideal Binary Clutters with a Triangle

Abdi, Ahmad; Guenin, Bertrand

doi:10.1007/s00493-018-3708-2

Cited by 2 publications

(2 citation statements)

References 22 publications

(25 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(See Figure 1.) The clutter of odd circuits of K 5 over its ten edges, denoted O 5 , is also binary, and it is non-ideal as 1 3 , 1 3 , . .…”

Section: Introductionmentioning

confidence: 99%

“…Let I, J be disjoint subsets of E. Denote by F \ I/J the clutter over E − (I ∪ J) of minimal sets of {C − J : C ∈ F, C ∩ I = ∅}. 1 We say that F \ I/J, and any clutter isomorphic to it, is a minor of F obtained after deleting I and contracting J. If I ∪ J = ∅, then F \ I/J is a proper minor of F. It is well-known that b(F \ I/J) = b(F)/I \ J [16].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Two-Point Fano and Ideal Binary Clutters

Abdi

Guenin

2019

Combinatorica

Self Cite

View full text Add to dashboard Cite

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 L 7 , O 5 , LC 7 as a minor. L 7 is the non-ideal clutter of the lines of the Fano plane, O 5 is the non-ideal clutter of odd circuits of the complete graph K 5 , and the two-point Fano LC 7 is the ideal clutter whose sets are the lines, and their complements, of the Fano plane that contain exactly one of two fixed points. In fact, we prove the following stronger statement: if F is a minimally non-ideal binary clutter different from L 7 , O 5 , b(O 5), then through every element, either F or b(F) has a two-point Fano minor.

show abstract

“…(See Figure 1.) The clutter of odd circuits of K 5 over its ten edges, denoted O 5 , is also binary, and it is non-ideal as 1 3 , 1 3 , . .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%