Discontinuity, Nonlinearity, and Complexity
A Note on the Connectivity of Binary Matroids
Discontinuity, Nonlinearity, and Complexity 11(3) (2022) 405408  DOI:10.5890/DNC.2022.09.004
Jun Yin$^{1,2,3}$, Bofeng Huo$^{4}$, HongJian 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), 305359],
Seymour introduced the binary matroid 3sums and proved that if a 3connected
binary matroid $M$ is a 3sum 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 3connected binary matroid $M$ expressed as a 3sum of
$M_1$ and $M_2$, we show that in general,
both $M_1$ and $M_2$ are 2connected, and
if $M_1$ and $M_2$ are simple matroids,
then both $M_1$ and $M_2$ are also 3connected.
References

[1]  Oxley, J.G. (2011), Matroid theory,
Oxford university Press, New York.


[2] 
Seymour, P.D. (1980), Decomposition of regular matroids, J. Combin.
Theory, Ser. B, 28 305359.


[3]  Seymour, P.D. (1981), Matroids and multicommodity flows,
European J. Combin. Theory Ser. B., 2 257290.


[4]  Bondy, J.A. and Murty, U.S.R. (2008), Graph Theory, Springer.
