Forbidden-minor characterization for the class of graphic element splitting matroids

Kiran Dalvi; Y.M. Borse; M.M. Shikare

Discussiones Mathematicae Graph Theory (2009)

  • Volume: 29, Issue: 3, page 629-644
  • ISSN: 2083-5892

Abstract

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This paper is based on the element splitting operation for binary matroids that was introduced by Azadi as a natural generalization of the corresponding operation in graphs. In this paper, we consider the problem of determining precisely which graphic matroids M have the property that the element splitting operation, by every pair of elements on M yields a graphic matroid. This problem is solved by proving that there is exactly one minor-minimal matroid that does not have this property.

How to cite

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Kiran Dalvi, Y.M. Borse, and M.M. Shikare. "Forbidden-minor characterization for the class of graphic element splitting matroids." Discussiones Mathematicae Graph Theory 29.3 (2009): 629-644. <http://eudml.org/doc/271019>.

@article{KiranDalvi2009,
abstract = {This paper is based on the element splitting operation for binary matroids that was introduced by Azadi as a natural generalization of the corresponding operation in graphs. In this paper, we consider the problem of determining precisely which graphic matroids M have the property that the element splitting operation, by every pair of elements on M yields a graphic matroid. This problem is solved by proving that there is exactly one minor-minimal matroid that does not have this property.},
author = {Kiran Dalvi, Y.M. Borse, M.M. Shikare},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {binary matroid; graphic matroid; minor; splitting operation; element splitting operation},
language = {eng},
number = {3},
pages = {629-644},
title = {Forbidden-minor characterization for the class of graphic element splitting matroids},
url = {http://eudml.org/doc/271019},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Kiran Dalvi
AU - Y.M. Borse
AU - M.M. Shikare
TI - Forbidden-minor characterization for the class of graphic element splitting matroids
JO - Discussiones Mathematicae Graph Theory
PY - 2009
VL - 29
IS - 3
SP - 629
EP - 644
AB - This paper is based on the element splitting operation for binary matroids that was introduced by Azadi as a natural generalization of the corresponding operation in graphs. In this paper, we consider the problem of determining precisely which graphic matroids M have the property that the element splitting operation, by every pair of elements on M yields a graphic matroid. This problem is solved by proving that there is exactly one minor-minimal matroid that does not have this property.
LA - eng
KW - binary matroid; graphic matroid; minor; splitting operation; element splitting operation
UR - http://eudml.org/doc/271019
ER -

References

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  1. [1] G. Azadi, Generalized splitting operation for binary matroids and related results (Ph.D. Thesis, University of Pune, 2001). 
  2. [2] Y.M. Borse, M.M. Shikare and Kiran Dalvi, Excluded-Minor characterization for the class of Cographic Splitting Matroids, Ars Combin., to appear. 
  3. [3] H. Fleischner, Eulerian Graphs and Related Topics, Part 1, Vol. 1 (North Holland, Amsterdam, 1990). Zbl0792.05091
  4. [4] A. Habib, Some new operations on matroids and related results (Ph.D. Thesis, University of Pune, 2005). 
  5. [5] F. Harary, Graph Theory (Addison-Wesley, Reading, 1969). 
  6. [6] J.G. Oxley, Matroid Theory (Oxford University Press, Oxford, 1992). 
  7. [7] T.T. Raghunathan, M.M. Shikare and B.N. Waphare, Splitting in a binary matroid, Discrete Math. 184 (1998) 267-271, doi: 10.1016/S0012-365X(97)00202-1. Zbl0955.05022
  8. [8] A. Recski, Matroid Theory and Its Applications (Springer Verlag, Berlin, 1989). 
  9. [9] M.M. Shikare and G. Azadi, Determination of the bases of a splitting matroid, European J. Combin. 24 (2003) 45-52, doi: 10.1016/S0195-6698(02)00135-X. Zbl1014.05018
  10. [10] M.M. Shikare, Splitting lemma for binary matroids, Southeast Asian Bull. Math. 32 (2007) 151-159. Zbl1199.05034
  11. [11] M.M. Shikare and B.N. Waphare, Excluded-Minors for the class of graphic splitting matroids, Ars Combin., to appear. Zbl1249.05048
  12. [12] P.J. Slater, A classification of 4-connected graphs, J. Combin. Theory 17 (1974) 281-298, doi: 10.1016/0095-8956(74)90034-3. Zbl0298.05136
  13. [13] W.T. Tutte, A theory of 3-connected graphs, Indag. Math. 23 (1961) 441-455. Zbl0101.40903
  14. [14] D.J.A. Welsh, Matroid Theory (Academic Press, London, 1976). 

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