A New Method of Checking the Consistency of Precedence Matrices
Journal of the ACM (JACM)
The organization of structured files
Communications of the ACM
Computer construction of minimal project networks
IBM Systems Journal
Szpilrajn's theorem on fuzzy orderings
Fuzzy Sets and Systems
On nilpotency of generalized fuzzy matrices
Fuzzy Sets and Systems
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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.