40,203 research outputs found
Towards a splitter theorem for internally 4-connected binary matroids VIII: small matroids
Our splitter theorem for internally 4-connected binary matroids studies pairs
of the form (M,N), where N and M are internally 4-connected binary matroids, M
has a proper N-minor, and if M' is an internally 4-connected matroid such that
M has a proper M'-minor and M' has an N-minor, then |E(M)|-|E(M')|>3. The
analysis in the splitter theorem requires the constraint that |E(M)|>15. In
this article, we complement that analysis by using an exhaustive computer
search to find all such pairs satisfying |E(M)|<16.Comment: Correcting minor error
Hypergraph polynomials and the Bernardi process
Recently O. Bernardi gave a formula for the Tutte polynomial of a
graph, based on spanning trees and activities just like the original
definition, but using a fixed ribbon structure to order the set of edges in a
different way for each tree. The interior polynomial is a generalization of
to hypergraphs. We supply a Bernardi-type description of using a
ribbon structure on the underlying bipartite graph . Our formula works
because it is determined by the Ehrhart polynomial of the root polytope of
in the same way as is. To prove this we interpret the Bernardi process as a
way of dissecting the root polytope into simplices, along with a shelling
order. We also show that our generalized Bernardi process gives a common
extension of bijections (and their inverses) constructed by Baker and Wang
between spanning trees and break divisors.Comment: 46 page
A version of Tutte's polynomial for hypergraphs
Tutte's dichromate T(x,y) is a well known graph invariant. Using the original
definition in terms of internal and external activities as our point of
departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. In
the definition, we associate activities to hypertrees, which are
generalizations of the indicator function of the edge set of a spanning tree.
We prove that hypertrees form a lattice polytope which is the set of bases in a
polymatroid. In fact, we extend our invariants to integer polymatroids as well.
We also examine hypergraphs that can be represented by planar bipartite graphs,
write their hypertree polytopes in the form of a determinant, and prove a
duality property that leads to an extension of Tutte's Tree Trinity Theorem.Comment: 49 page
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