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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


A Note on the Connectivity of Binary Matroids

Discontinuity, Nonlinearity, and Complexity 11(3) (2022) 405--408 | DOI:10.5890/DNC.2022.09.004

Jun Yin$^{1,2,3}$, Bofeng Huo$^{4}$, Hong-Jian Lai$^{5}$

$^1$ School of Computer, Qinghai Normal University, Xining, Qinghai, 810008, P.R. of China

$^2$ Key Laboratory of Tibetan Information Processing and Machine Translation, Qinghai Province

$^3$ Key Laboratory of Tibetan Information Processing, Ministry of Education

$^4$ School of Mathematics and Statistics, Qinghai Normal University, Xining, Qinghai 810016, PRC

$^5$ Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA

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Abstract

In [J. Combinatorial Theory, Ser. B, 28 (1980), 305-359], Seymour introduced the binary matroid 3-sums and proved that if a 3-connected binary matroid $M$ is a 3-sum of matroids $M_1$ and $M_2$, then each of $M_1$ and $M_2$ is isomorphic to a proper minor of $M$. For a 3-connected binary matroid $M$ expressed as a 3-sum of $M_1$ and $M_2$, we show that in general, both $M_1$ and $M_2$ are 2-connected, and if $M_1$ and $M_2$ are simple matroids, then both $M_1$ and $M_2$ are also 3-connected.

References

  1. [1]  Oxley, J.G. (2011), Matroid theory, Oxford university Press, New York.
  2. [2]  Seymour, P.D. (1980), Decomposition of regular matroids, J. Combin. Theory, Ser. B, 28 305-359.
  3. [3]  Seymour, P.D. (1981), Matroids and multicommodity flows, European J. Combin. Theory Ser. B., 2 257-290.
  4. [4]  Bondy, J.A. and Murty, U.S.R. (2008), Graph Theory, Springer.