A NOTE ON CONNECTIVITY PRESERVING SPLITTING OPERATION FOR MATROIDS REPRESENTABLE OVER $GF(p)$
Print ISSN: 0972-7752 | Online ISSN: 2582-0850 | Total Downloads : 13
DOI: https://doi.org/10.56827/SEAJMMS.2024.2001.20
Author :
Sachin Gunjal (Department of Mathematics and Statistics, Dr. Vishwanath Karad MIT World Peace University, Pune - 411038, Maharashtra, INDIA)
Prashant Malavadkar (Department of Mathematics and Statistics, Dr. Vishwanath Karad MIT World Peace University, Pune - 411038, Maharashtra, INDIA)
Uday Jagadale (Department of Mathematics and Statistics, Dr. Vishwanath Karad MIT World Peace University, Pune - 411038, Maharashtra, INDIA)
Abstract
The splitting operation on a $p$-matroid does not necessarily preserve connectivity. It is observed that there exists a single element extension of the splitting matroid which is connected. In this paper, we define the element splitting operation on $p$-matroids which consist of a splitting operation followed by a single element extension. It is proved that the element splitting operation on a connected $p$-matroid yields a connected $p$-matroid. We give a sufficient condition to yield Eulerian $p$-matroid from Eulerian $p$-matroid under the element splitting operation. A sufficient condition to obtain Hamiltonian $p$-matroid by applying the element splitting operation on $p$-matroid is also provided. The characterization of the paving $p$-matroid which are closed under the element splitting operation, is also obtained.
Keywords and Phrases
$p$-matroid, element splitting operation, Eulerian matroid, connected matroid, hamiltonian matroid, elementary lift, paving matroid.
A.M.S. subject classification
05B35, 05C50, 05C83.
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