# Topoligical classification of $\Omega$-stable flows on surfaces by means of effectively distinguishable multigraphs

## TL;DR

This paper develops a combinatorial topological classification of $\, ext{Ω}$-stable flows on surfaces using multigraphs, providing polynomial-time algorithms for distinguishing, orientability, and computing the Euler characteristic.

## Contribution

It introduces a complete topological invariant for $\, ext{Ω}$-stable flows based on multigraphs and offers efficient algorithms for their classification and properties analysis.

## Key findings

- Complete topological invariant via multigraphs for $\, ext{Ω}$-stable flows
- Polynomial-time algorithms for graph isomorphism and orientability
- Graph-based formula for Euler characteristic

## Abstract

Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads to $\Omega$-stable flows on surfaces, which are not structurally stable. However, in the present paper we prove that a topological classification of such flows is also reduced to a combinatorial problem. Our complete topological invariant is a multigraph, and we present a polynomial-time algorithm for the distinction of such graphs up to an isomorphism. We also present a graph criterion for orientability of the ambient manifold and a graph-associated formula for its Euler characteristic. Additionally, we give polynomial-time algorithms for checking the orientability and calculating the characteristic.