Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the representability of totally unimodular matrices on bidirected
  graphs   Seymour's famous decomposition theorem for regular matroids states that any
totally unimodular (TU) matrix can be constructed through a series of
composition operations called $k$-sums starting from network matrices and their
transposes and two compact representation matrices $B_{1}, B_{2}$ of a certain
ten element matroid. Given that $B_{1}, B_{2}$ are binet matrices we examine
the $k$-sums of network and binet matrices. It is shown that the $k$-sum of a
network and a binet matrix is a binet matrix, but binet matrices are not closed
under this operation for $k=2,3$. A new class of matrices is introduced the so
called {\em tour matrices}, which generalises network, binet and totally
unimodular matrices. For any such matrix there exists a bidirected graph such
that the columns represent a collection of closed tours in the graph. It is
shown that tour matrices are closed under $k$-sums, as well as under pivoting
and other elementary operations on its rows and columns. Given the constructive
proofs of the above results regarding the $k$-sum operation and existing
recognition algorithms for network and binet matrices, an algorithm is
presented which constructs a bidirected graph for any TU matrix.

24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org