On the Compatibility Between a Graph and a Simple Order.

by Fred S. Roberts

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A formal graph-theoretic approach to measurement problems, such as those occurring in psychology, sociology, and economics. The relation of indifference is represented by adjacency in a graph. Since indifference is symmetric, the graph is unoriented (preference relations are asymmetric and result in oriented graphs). The psychological notion of threshold, or just noticeable difference (jnd), suggests a problem in which numbers are assigned to points so that indifference exists relative to the two objects represented (i.e., so that the two points are adjacent) if and only if the numbers are close. The present study generalizes the problem to that of ordering the points of a graph (rather than numbering them) so that two nonadjacent points cannot be included between two adjacent points. This notion of compatible ordering turns out to be more general than numbering because it permits theorems to be proved for graphs with an arbitrarily large (infinite) number of points. Criteria are found for determining whether the points of an unoriented graph (or an oriented graph) can be ordered in a way compatible with the adjacency relation, and it is shown that if such an order exists, it is essentially unique. Criteria are then given for the numerical assignment problem in the case where the jnds vary. Techniques used include formal logic. 36 pp. Refs.

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