Graphs and Combinatorics

, Volume 34, Issue 3, pp 427–441 | Cite as

On Binary Matroids Without a \(P_{10}\)-Minor

  • Xiangqian Zhou
Original Paper


We study the class of binary matroids without a \(P_{10}\)-minor and find all internally 4-connected non-regular matroids in the class.


Binary matroids 3-connected Internally 4-connected \(P_{10}\) 

Mathematics Subject Classification



  1. 1.
    Bouchet, A., Cunningham, W.H., Geelen, J.F.: Principle unimodular skew-symmetric matrices. Combinatorica 18, 461–486 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ding, G., Wu, H.: Characterizing binary matroids with no \(P_9\)-minor. Adv. Appl. Math. 70, 70–91 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Geelen, J.F., Gerards, A.M.H., Kapoor, A.: The excluded minors for \(GF(4)\)-representable matroids. J. Combin. Theory, Ser. B 9, 247–299 (2001)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hlinĕný, P.: The Macek Program. (2005)
  5. 5.
    Kingan, S.R.: A generalization of a graph result of D.W. Hall. Discrete Math. 173, 129–135 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kingan, S.R., Lemos, M.: A decomposition theorem for binary matroids with no prism minor. Graphs Comb. 30(6), 1479–1497 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kingan, S.R., Lemos, M.: Strong splitter theorem. Ann. Comb. 18, 111–116 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mayhew, D., Royle, G.: The internally 4-connected binary matroids with no \(M(K_{5}\ e)\)- minor. SIAM J. Discrete Math. 26(2), 755–767 (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Mayhew, D., Royle, G., Whittle, G.: The Internally 4-Connected Binary Matroids with no \(M(K_{3,3})\)-Minor, Memoirs of the American Mathematical Society, vol. 981. American Mathematical Society, Providence (2011)zbMATHGoogle Scholar
  10. 10.
    Oxley, J.G.: The binary matroids with no 4-wheel minor. Trans. Am. Math. Soc. 301, 63–75 (1987)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Oxley, J.G.: Matroid Theory. Oxford University Press, New York (1992)zbMATHGoogle Scholar
  12. 12.
    Seymour, P.D.: Decomposition of regular matroids. J. Combin. Theory Ser. B 28, 305–359 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Williams, J.T., Zhou, X.: A new proof for a result of Kingan and Lemos. Graphs Comb. 32(1), 403–417 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Qin, H., Zhou, X.: The class of binary matroids with no \(M(K_{3,3})\)-, \(M^{*}(K_{3,3})\)-, \(M(K_5)\)-, or \(M^{*}(K_5)\)-minor. J. Comb. Theory Ser. B 90, 173–184 (2004)CrossRefzbMATHGoogle Scholar
  15. 15.
    Zhou, X.: On internally 4-connected non-regular binary matroids. J. Comb. Theory Ser. B 91, 327–343 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsWright State UniversityDaytonUSA

Personalised recommendations