Many problems in program optimizationn have been solved by applying a technique called interval analysis to the flow graph of the program. A flow graph which is susceptible to this type of analysis is called reducible. This paper describes an algorithm for testing whether a flow graph is reducible. The algorithm uses depth-first search to reveal the structure of the flow graph and a good method for computing disjoint set unions to determine reducibility from the search information. When the algorithm is implemented on a random access computer, it requires $O(E \log^{} E)$ time to analyze a graph with E edges, where $log^{} x=\min{i|\log^{i}x \leq 1}$. The time bound compares favorably with the $O(E \log E)$ bound of a previously known algorithm.