A graph traversal system is described herein that efficiently identifies strongly connected components with entries, exits, and corresponding edges at the same time. Entry and exit nodes can be recognized by scanning every node after the strongly connected components have been identified, but revisiting these nodes incurs undesirable overhead. The graph traversal system identifies entries and exits during a single pass while the strongly connected components are being identified. In addition, the system modifies the semantics for some applications so that a single node all alone is not considered to be a strongly connected component. Thus, the graph traversal system allows efficient identification of entries to and exits from strongly connected components in a manner that can be applied to a variety of computer software problems that use directed graphs for data structures.