38 research outputs found
Triangle-roundedness in matroids
A matroid is said to be triangle-rounded in a class of matroids
if each -connected matroid with a triangle
and an -minor has an -minor with as triangle. Reid gave a result
useful to identify such matroids as stated next: suppose that is a binary
-connected matroid with a -connected minor , is a triangle of
and ; then has a -connected minor with an
-minor such that is a triangle of and . We
strengthen this result by dropping the condition that such element exists
and proving that there is a -connected minor of with an -minor
such that is a triangle of and . This
result is extended to the non-binary case and, as an application, we prove that
is triangle-rounded in the class of the regular matroids
On two classes of nearly binary matroids
We give an excluded-minor characterization for the class of matroids M in
which M\e or M/e is binary for all e in E(M). This class is closely related to
the class of matroids in which every member is binary or can be obtained from a
binary matroid by relaxing a circuit-hyperplane. We also provide an
excluded-minor characterization for the second class.Comment: 14 pages, 4 figures. This paper has been accepted for publication in
the European Journal of Combinatorics. This is the final version of the pape
A Characterization of Certain Excluded-Minor Classes of Matroids
A result of Walton and the author establishes that every 3-connected matroid of rank and corank at least three has one of five six-element rank-3 self-dual matroids as a minor. This paper characterizes two classes of matroids that arise when one excludes as minors three of these five matroids. One of these results extends the author\u27s characterization of the ternary matroids with no M(K4)-minor, while the other extends Tutte\u27s excluded-minor characterization of binary matroids. © 1989, Academic Press Limited. All rights reserved
A notion of minor-based matroid connectivity
For a matroid , a matroid is -connected if every two elements of
are in an -minor together. Thus a matroid is connected if and only if it
is -connected. This paper proves that is the only connected
matroid such that if is -connected with , then or is -connected for all elements . Moreover, we
show that and are the only connected matroids
such that, whenever a matroid has an -minor using and an -minor
using , it also has an -minor using . Finally, we show
that is -connected if and only if every clonal
class of is trivial.Comment: 13 page
On Density-Critical Matroids
For a matroid having rank-one flats, the density is
unless , in which case . A matroid is
density-critical if all of its proper minors of non-zero rank have lower
density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among
simple matroids that cannot be covered by independent sets is
density-critical. It is straightforward to show that is the only
minor-minimal loopless matroid with no covering by independent sets. We
prove that there are exactly ten minor-minimal simple obstructions to a matroid
being able to be covered by two independent sets. These ten matroids are
precisely the density-critical matroids such that but for all proper minors of . All density-critical matroids of density
less than are series-parallel networks. For , although finding all
density-critical matroids of density at most does not seem straightforward,
we do solve this problem for .Comment: 16 page