# Resolution of Indecomposable Integral Flows on Signed Graphs

## TL;DR

This paper characterizes indecomposable integral flows on signed graphs, revealing their complexity beyond unsigned graphs through a novel approach involving double covering graphs and singularity resolution.

## Contribution

It provides a complete description of indecomposable flows on signed graphs using a new method based on resolving singularities with double covering graphs.

## Key findings

- Indecomposable flows on signed graphs are more complex than on unsigned graphs.
- A new method using double covering graphs is introduced for analyzing these flows.
- The paper offers a comprehensive classification of indecomposable flows on signed graphs.

## Abstract

It is well known that each nonnegative integral flow on a graph can be decomposed into a sum of nonnegative graphic circuit flows, which cannot be further decomposed into nonnegative integral sub-flows. This is equivalent to saying that the indecomposable flows on graphs are those graphic circuit flows. Turning from graphs to signed graphs, the indecomposable flows are much richer than those of unsigned graphs. This paper gives a complete description of indecomposable flows on signed graphs from the viewpoint of resolution of singularities by means of double covering graphs.