  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.
