ABSTRACT. A procedure for determining whether two graphs are isomorphic is described. During the procedure, from any given graph two graphs, the representative graph and the reordered graph, are derived.The representative graph is a homomorphic image of the original graph; the reordered graph is constructed from the representative graph to be isomorphic to the given graph. Unique labels are assigned to the vertices of both derived graphs. It follows that two representative graphs or two reordered graphs are isomorphic if and only if they are identical. A conjecture states that the representative graphs exhibit the automorphism partitioning of the given graph. The representative graphs form a necessity condition for isomorphism; namely, if the representative graphs are not identical, then the given graphs are not isomorphic. The converse is true for trees and follows from the conjecture for other types of graphs. It is also shown that the reordered graphs form a sufficiency condition for isomorphism; namely, if the reordered graphs are identical, then the given graphs are isomorphic. The converse follows from the conjecture.The time required to determine both derived graphs depends on a power of n, the order of the given graph. This power is a function of an adjacency property known as the strong regularity of the given graph. For graphs that do not contain a strongly regular transitive subgraph, the power is, at worst, five.

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A unifying model for the study of database performance is proposed. Applications of the model are shown to relate and extend important work concerning batched searching, transposed files, index selection, dynamic hash-based files, generalized access path structures, differential files, network databases, and multifile query processing.

A procedure for constructing a minimal event-node network to represent a set of precedence relations without parallel activities is presented. A minimal eventnode network is an event-node network in which both the number of nodes and the number of arcs are the minima to preserve the given precedence relations. Counterexamples are given to show that the algorithm presented by A.C. Fisher, J.S. Liebman, and G.L. Nemhauser (1968) produces event-node networks which are not minimal. Since our procedure includes the set-covering problem, the time required may grow exponentially with the number of given activities.