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A new algorithm is presented for detecting graph monomorphisms for a pair of graphs. This algorithm entails a tree search based on the projections of the product graph called the net of the two graphs. It uses the minimum number of neighbors of the projected graphs to detect infeasible subtrees. The algorithm, in comparison with that of Deo and coworkers, is more efficient in its storage space utilization and average execution time. It does not suffer from the ambiguity which arises in Deo et al.'s work when cyclic graphs are matched. Applications to attributed graph monomorphisms are included.