The Two-Point Fano and Ideal Binary Clutters


Let \(\mathbb{F}\) be a binary clutter. We prove that if \(\mathbb{F}\) is non-ideal, then either \(\mathbb{F}\) or its blocker \(b(\mathbb{F})\) has one of \(\mathbb{L}_7,\mathbb{O}_5,\mathbb{LC}_7\) as a minor. \(\mathbb{L}_7\) is the non-ideal clutter of the lines of the Fano plane, \(\mathbb{O}_5\) is the non-ideal clutter of odd circuits of the complete graph K5, and the two-point Fano\(\mathbb{LC}_7\) 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 \(\mathbb{F}\) is a minimally non-ideal binary clutter different from \(\mathbb{L}_7,\mathbb{O}_5,b(\mathbb{O}_5)\), then through every element, either \(\mathbb{F}\) or \(b(\mathbb{F})\) has a two-point Fano minor.

This is a preview of subscription content, access via your institution.


  1. [1]

    A. Abdi and B. Guenin: The minimally non-ideal binary clutters with a triangle, submitted.

  2. [2]

    W. G. Bridges and H. J. Ryser: Combinatorial designs and related systems, J. Algebra 13 (1969), 432–446

    MathSciNet  Article  Google Scholar 

  3. [3]

    G. Cornuéjols: Combinatorial optimization, packing and covering, SIAM, Philadelphia, (2001).

    Google Scholar 

  4. [4]

    G. Cornuéjols and B. Guenin: Ideal binary clutters, connectivity, and a conjecture of Seymour, SIAM J. Discrete Math. 15 (2002), 329–352.

    MathSciNet  Article  Google Scholar 

  5. [5]

    J. Edmonds and D. R. Fulkerson: Bottleneck extrema, J. Combin. Theory Ser. B 8 (1970), 299–306.

    MathSciNet  Article  Google Scholar 

  6. [6]

    L. R. Ford and D. R. Fulkerson: Maximal flow through a network, Canadian J. Math. 8 (1956), 399–404.

    MathSciNet  Article  Google Scholar 

  7. [7]

    B. Guenin: A characterization of weakly bipartite graphs, J. Combin. Theory Ser. B 83 (2001), 112–168.

    MathSciNet  Article  Google Scholar 

  8. [8]

    B. Guenin: Integral polyhedra related to even-cycle and even-cut matroids, Math. Oper. Res. 27 (2002), 693–710.

    MathSciNet  Article  Google Scholar 

  9. [9]

    A. Lehman: A solution of the Shannon switching game, Society for Industrial Appl. Math. 12 (1964), 687–725.

    MathSciNet  Article  Google Scholar 

  10. [10]

    A. Lehman: On the width-length inequality, Math. Program. 17 (1979), 403–417.

    MathSciNet  Article  Google Scholar 

  11. [11]

    A. Lehman: The width-length inequality and degenerate projective planes, DIMACS Vol. 1 (1990), 101–105.

    MathSciNet  MATH  Google Scholar 

  12. [12]

    K. Menger: Zur allgemeinen Kurventheorie, Fundamenta Mathematicae 10 (1927), 96–115.

    Article  Google Scholar 

  13. [13]

    B. Novick and A. Sebő: On combinatorial properties of binary spaces, IPCO 4 (1995), 212–227.

    MathSciNet  Google Scholar 

  14. [14]

    J. Oxley: Matroid theory, second edition, Oxford University Press, New York, 2011.

    Google Scholar 

  15. [15]

    P. D. Seymour: On Lehman's width-length characterization, DIMACS 1 (1990), 107–117.

    MathSciNet  MATH  Google Scholar 

  16. [16]

    P. D. Seymour: The forbidden minors of binary matrices, J. London Math. Society 2 (1976), 356–360.

    Article  Google Scholar 

  17. [17]

    P. D. Seymour: The matroids with the maxow min-cut property, J. Combin. Theory Ser. B 23 (1977), 189–222.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Ahmad Abdi.

Additional information

This work is supported by NSERC CGS and Discovery grants and by U.S. Office of Naval Research grants under award numbers N00014-15-1-2171 and N00014-18-1-2078.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Abdi, A., Guenin, B. The Two-Point Fano and Ideal Binary Clutters. Combinatorica 39, 753–777 (2019).

Download citation

Mathematics Subject Classification (2010)

  • 90C57
  • 05B35