$k$-Sum Decomposition of Strongly Unimodular Matrices

TL;DR

This paper explores the algebraic structure of networks using matroid theory, demonstrating that strongly unimodular matrices can be decomposed into connected blocks via $k$-sums, revealing new structural insights.

Contribution

It establishes that strongly unimodular matrices are closed under $k$-sums for $k=1,2$, enabling their decomposition into highly connected network-representing blocks.

Findings

Strongly unimodular matrices are closed under $k$-sums for $k=1,2

Decomposition results connect matroid theory to network matrices

Networks can be studied through their matrix decompositions

Abstract

Networks are frequently studied algebraically through matrices. In this work, we show that networks may be studied in a more abstract level using results from the theory of matroids by establishing connections to networks by decomposition results of matroids. First, we present the implications of the decomposition of regular matroids to networks and related classes of matrices, and secondly we show that strongly unimodular matrices are closed under $k$-sums for $k=1,2$ implying a decomposition into highly connected network-representing blocks, which are also shown to have a special structure.