Geodesic
Forbidden-minor characterization for the class of graphic element splitting matroids
Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 3, pp. 629-644

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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.
Keywords: binary matroid, graphic matroid, minor, splitting operation, element splitting operation 
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     title = {Forbidden-minor characterization for the class of graphic element splitting matroids},
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Dalvi, Kiran; Borse, M.; Shikare, M. Forbidden-minor characterization for the class of graphic element splitting matroids. Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 3, pp. 629-644. http://geodesic.mathdoc.fr/item/DMGT_2009_29_3_a12/