Theory and applications of freedom in matroids.
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.