Generalization of pinching operation to binary matroids

Year 2020, Volume: 7 Issue: 3, 247 - 258, 06.09.2020

Vahid Ghorbani Ghodratollah Azadi Habib Azanchiler

https://doi.org/10.13069/jacodesmath.784992

Abstract

In this paper, we generalize the pinching operation on two edges of graphs to binary
matroids and investigate some of its basic properties. For $n\geq 2$, the matroid that is obtained from an $n$-connected matroid by this operation is a $k$-connected matroid with $k\in\{2,3,4\}$ or is a disconnected matroid. We find conditions to guarantee this $k$. Moreover, we show that Eulerian binary matroids are characterized by this operation and we also provide some interesting applications of this operation.

Keywords

Binary matroid, Connectivity, Pinching, Splitting, Splitting off, Element splitting

References

• [1] G. Azadi, Generalized splitting operation for binary matroids and related results, Ph.D. Thesis, University of Pune, 2001.
• [2] A. Frank, Edge–connection of graphs, digraphs, and hypergraphs, More Sets, Graphs and Numbers 15 (2006) 93–141.
• [3] M. M. Shikare, K. V. Dalvi, S. B. Dhotre, Splitting off operation for binary matroids and its applications, Graph and Combinatorics 27 (2011) 871–882.
• [4] J. Oxley, Matroid Theory, Oxford University Press, 2nd ed. 2011.
• [5] T. T. Raghunathan, M. M. Shikare, B. N. Waphare, Splitting in a binary matroid, Discrete Math. 184 (1998) 267–271.
• [6] D. J. A. Welsh, Euler and bipartite matroids, Journal of Combinatorial Theory 6(4) (1969) 375–377.
• [7] D. West, Introduction to graph theory, Prentice–Hall, 2nd ed. 2001.