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Let $(S,\Phi)$ be a pair of a closed oriented surface and $\Phi$ be a realanalytic flow with finitely many singularities. Let $x$ be a point of $S$ withthe polycycle $\omega$-limit set $\omega(x)$. In this paper we give topologicalclassification of $\omega(x)$. Our main theorem says that $\omega(x)$ isdiffeomorphic to the boundary of a cactus in the $2$-sphere $S^{2}$. Moreover$S$ is a connected sum of the above $S^{2}$ and a closed oriented surface alongfinitely many embedded circles which are disjoint from $\omega(x)$. This givesa natural generalization to the higher genus of the main result of \cite{JL}for the genus $0$ case. Our result is further applicable to a larger class ofsurface flows, a compact oriented surface with corner and a $C^{1}$-flow withfinitely many singularities locally diffeomorphic to an analytic flow.

Polycycle omega-limit sets of flows on the compact Riemann surfaces and Eulerian path