On the representability of totally unimodular matrices on bidirected graphs

TL;DR

This paper explores the structure of totally unimodular matrices through bidirected graphs, introducing tour matrices as a new class and providing an algorithm to construct bidirected graphs for any TU matrix.

Contribution

It introduces tour matrices, generalizes the class of matrices representable by bidirected graphs, and presents an algorithm for constructing such graphs from any TU matrix.

Findings

The k-sum of a network and a binet matrix is a binet matrix.

Binet matrices are not closed under k-sums for k=2,3.

An algorithm is provided to construct bidirected graphs for any TU matrix.

Abstract

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.