Theory and applications of freedom in matroids

• Main Author: Duke, R.
• Published: Open University 1981
• Subjects: 510; Pure mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.354270

To each cell e in a matroid M we can associate a non-negative integer lIell called the freedom of e. Geometrically the value Ilell indicates how freely placed the cell ~s ~n the matroid. We see tha t II e II ~s equal to the degree of the modular cut generated by all the fully-dependent flats of M containing e . The relationship between freedom and basic matroid constructions, particularly one-point lifts and duality, is examined, and then applied to erections. We see that the number of times a matroid M can be erected is related to the degree of the modular cut generated by all the fully-dependent flats of W<. If Z;;(M) is the set of integer polymatroids with underlying matroid structure M, then we show that for any cell e of M II ell max f (e) • f E 1;; (M) We look at freedom in binary matroids and show that for a connected binary matroid M, II e II is the number of connec ted components of M\e. Finally the matroid join is examined and we are able to solve a conjecture of Lovasz and Recski that a connected binary matroid M is reducible if and only if there is a cell e of M with M\e disconnected.