On the Consistency of Precedence Matrices

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Abstract
The consistency of precedence matrices is studied in the very natural geometric setting of the theory of directed graphs. An elegant recent procedure (Marimont [7]) for checking consistency is further justified by means of a graphical lemma. In addition, the “direction of future work” mentioned in [7] (to which the present communication may be regarded as a sequel) is developed here using graph theoretic methods. This is based on the relationship between the occurrence of directed cycles and the recognition of “strongly connected components” in a directed graph. An algorithm is included for finding these components in any directed graph. This is necessarily more complicated than determining whether there do not exist any directed cycles, i.e., whether or not a given precedence matrix is consistent.

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