Classification of Indecomposable Flows of Signed Graphs

TL;DR

This paper classifies indecomposable flows in signed graphs, revealing their structure as characteristic vectors of circuits and Eulerian cycle-trees, and shows their richer variety compared to unsigned graphs.

Contribution

It introduces a classification of indecomposable flows in signed graphs into circuits and Eulerian cycle-trees, expanding understanding of their structure.

Findings

Indecomposable flows correspond to circuit and Eulerian cycle-tree characteristic vectors.

Non-circuit indecomposable flows can be decomposed into sums of half circuit characteristic vectors.

The variety of indecomposable flows in signed graphs exceeds that of unsigned graphs.

Abstract

An indecomposable flow $f$ on a signed graph $\Sigma$ is a nontrivial integral flow that cannot be decomposed into $f=f_1+f_2$, where $f_1,f_2$ are nontrivial integral flows having the same sign (both $\geq 0$ or both $\leq 0$) at each edge of $\Sigma$. This paper is to classify indecomposable flows into characteristic vectors of circuits and Eulerian cycle-trees --- a class of signed graphs having a kind of tree structure in which all cycles can be viewed as vertices of a tree. Moreover, each indecomposable flow other than circuit characteristic vectors can be further decomposed into a sum of certain half circuit characteristic vectors having the same sign at each edge. The variety of indecomposable flows of signed graphs is much richer than that of ordinary unsigned graphs.