We propose a new efficient algorithm for solving the cluster labeling problem in support vector clustering (SVC). The proposed algorithm analyzes the topology of the function describing the SVC cluster contours and explores interconnection paths between critical points separating distinct cluster contours. This process allows distinguishing disjoint clusters and associating each point to its respective one. The proposed algorithm implements a new fast method for detecting and classifying critical points while analyzing the interconnection patterns between them. Experiments indicate that the proposed algorithm significantly improves the accuracy of the SVC labeling process in the presence of clusters of complex shape, while reducing the processing time required by existing SVC labeling algorithms by orders of magnitude.