2,204 research outputs found
Infinite matroids in graphs
It has recently been shown that infinite matroids can be axiomatized in a way
that is very similar to finite matroids and permits duality. This was
previously thought impossible, since finitary infinite matroids must have
non-finitary duals. In this paper we illustrate the new theory by exhibiting
its implications for the cycle and bond matroids of infinite graphs. We also
describe their algebraic cycle matroids, those whose circuits are the finite
cycles and double rays, and determine their duals. Finally, we give a
sufficient condition for a matroid to be representable in a sense adapted to
infinite matroids. Which graphic matroids are representable in this sense
remains an open question.Comment: Figure correcte
Matroid and Tutte-connectivity in infinite graphs
We relate matroid connectivity to Tutte-connectivity in an infinite graph.
Moreover, we show that the two cycle matroids, the finite-cycle matroid and the
cycle matroid, in which also infinite cycles are taken into account, have the
same connectivity function. As an application we re-prove that, also for
infinite graphs, Tutte-connectivity is invariant under taking dual graphs.Comment: 11 page
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
A decomposition theorem for binary matroids with no prism minor
The prism graph is the dual of the complete graph on five vertices with an
edge deleted, . In this paper we determine the class of binary
matroids with no prism minor. The motivation for this problem is the 1963
result by Dirac where he identified the simple 3-connected graphs with no minor
isomorphic to the prism graph. We prove that besides Dirac's infinite families
of graphs and four infinite families of non-regular matroids determined by
Oxley, there are only three possibilities for a matroid in this class: it is
isomorphic to the dual of the generalized parallel connection of with
itself across a triangle with an element of the triangle deleted; it's rank is
bounded by 5; or it admits a non-minimal exact 3-separation induced by the
3-separation in . Since the prism graph has rank 5, the class has to
contain the binary projective geometries of rank 3 and 4, and ,
respectively. We show that there is just one rank 5 extremal matroid in the
class. It has 17 elements and is an extension of , the unique splitter
for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's
result identifying the binary internally 4-connected matroids with no prism
minor [5]
Axioms for infinite matroids
We give axiomatic foundations for non-finitary infinite matroids with
duality, in terms of independent sets, bases, circuits, closure and rank. This
completes the solution to a problem of Rado of 1966.Comment: 33 pp., 2 fig
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