On the separatrix graph of a rational vector field on the Riemann sphere (Q2656253)

The authors consider the flow on the Riemann sphere determined by a complex rational vector field. This flow has no limit cycles. Therefore the global behavior of the phase curves is determined by the separatrices and a separatrix graph is constructed. The notion of an admissible graph is introduced. It is proved that a planar directed graph is homeomorphic to the separatrix graph of a rational vector field if and only if it is an admissible graph.