Classes of Matroids

Abstract

This paper explores which classes of graphs and matroids are k-balanced. A connection between k-balanced graphs and k-balanced matroids was also obtained. In this paper, we continue our study of the class of k-balanced matroids in order to see what matroid operations preserve k-balance. Since strong maps of matroids are defined as analogues of continuous maps of topological spaces, it is natural to ask what other topological notions carry over to matroids. In characterizing strong maps from 2000 to 2003, Al-Hawary defined a closure matroid to be a matroid in which $\overline{A\cup B}=\overline{A}\cup \overline{B}$ for all subsets $A$ and $B$ of its ground set. We obtain a new classification of closure matroids. Moreover, necessary and sufficient conditions for the direct sum, parallel extension connection and series extension connection to preserve k-balance property are given. Keywords: Amalgam; Balance; K-density; Graph; Matroid; Closure matroid.