4,206 research outputs found
Nullity and Loop Complementation for Delta-Matroids
We show that the symmetric difference distance measure for set systems, and
more specifically for delta-matroids, corresponds to the notion of nullity for
symmetric and skew-symmetric matrices. In particular, as graphs (i.e.,
symmetric matrices over GF(2)) may be seen as a special class of
delta-matroids, this distance measure generalizes the notion of nullity in this
case. We characterize delta-matroids in terms of equicardinality of minimal
sets with respect to inclusion (in addition we obtain similar characterizations
for matroids). In this way, we find that, e.g., the delta-matroids obtained
after loop complementation and after pivot on a single element together with
the original delta-matroid fulfill the property that two of them have equal
"null space" while the third has a larger dimension.Comment: Changes w.r.t. v4: different style, Section 8 is extended, and in
addition a few small changes are made in the rest of the paper. 15 pages, no
figure
Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices
We prove that every infinite sequence of skew-symmetric or symmetric matrices
M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such
that M_i is isomorphic to a principal submatrix of the Schur complement of a
nonsingular principal submatrix in M_j, if those matrices have bounded
rank-width. This generalizes three theorems on well-quasi-ordering of graphs or
matroids admitting good tree-like decompositions; (1) Robertson and Seymour's
theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's
theorem for matroids representable over a fixed finite field having bounded
branch-width, and (3) Oum's theorem for graphs of bounded rank-width with
respect to pivot-minors.Comment: 43 page
Circuit-Difference Matroids
One characterization of binary matroids is that the symmetric difference of
every pair of intersecting circuits is a disjoint union of circuits. This paper
considers circuit-difference matroids, that is, those matroids in which the
symmetric difference of every pair of intersecting circuits is a single
circuit. Our main result shows that a connected regular matroid is
circuit-difference if and only if it contains no pair of skew circuits. Using a
result of Pfeil, this enables us to explicitly determine all regular
circuit-difference matroids. The class of circuit-difference matroids is not
closed under minors, but it is closed under series minors. We characterize the
infinitely many excluded series minors for the class.Comment: 11 page
Matroidal structure of generalized rough sets based on symmetric and transitive relations
Rough sets are efficient for data pre-process in data mining. Lower and upper
approximations are two core concepts of rough sets. This paper studies
generalized rough sets based on symmetric and transitive relations from the
operator-oriented view by matroidal approaches. We firstly construct a
matroidal structure of generalized rough sets based on symmetric and transitive
relations, and provide an approach to study the matroid induced by a symmetric
and transitive relation. Secondly, this paper establishes a close relationship
between matroids and generalized rough sets. Approximation quality and
roughness of generalized rough sets can be computed by the circuit of matroid
theory. At last, a symmetric and transitive relation can be constructed by a
matroid with some special properties.Comment: 5 page
Tableaux in the Whitney Module of a Matroid
The Whitney module of a matroid is a natural analogue of the tensor algebra
of the exterior algebra of a vector space that takes into account the
dependencies of a matroid. In this paper we indicate the role that tableaux can
play in describing the Whitney module. We will use our results to describe a
basis of the Whitney module of a certain class of matroids known as freedom
matroids (also known as Schubert, or shifted matroids). The doubly multilinear
submodule of the Whitney module is a representation of the symmetric group. We
will describe a formula for the multiplicity of hook shapes in this
representation in terms of no broken circuit sets
Lagrangian Matroids: Representations of Type
We introduce the concept of orientation for Lagrangian matroids represented
in the flag variety of maximal isotropic subspaces of dimension N in the real
vector space of dimension 2N+1. The paper continues the study started in
math.CO/0209100.Comment: Requires amssymb.sty; 17 page
- …