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For the given directed graphs, a method based on Moore sequential machines (MSM) for solving the graph isomorphism problem was recently proposed. However, that method provides only a necessary (but not a sufficient) condition. This correspondence corrects this error by developing a necessary and sufficient condition. It will be shown that this type of problem can be also solved by using deterministic state machines (DSM). But there are tradeoffs to be considered in using either DSM's or MSM's. In addition, the effectiveness of the method based on MSM's depends very much on the output definition.