Classification and specification of flat cluster methods

Abstract Jardine and Sibson's flat cluster methods exhibit mathematical properties that make them particularly appropriate for use in taxonomy. But since the number of flat cluster methods increases at least exponentially with the number of objects to be classified, it seems desirable to classify flat cluster methods, so as to isolate subclasses appropriate to the construction of taxonomic systems, and to develop an effective mechanism for specifying flat cluster methods. In this paper a classification scheme is developed that is based on characteristics of confinement, consistency, stability, and authenticity. A type of graph-theoretic property, the normal property, is described, and it is shown by means of simple transformations that every normal property specifies a flat cluster method and that each flat cluster method is specified by at least one normal property. To illustrate these results, ten clustering properties described by Hubert are used to construct normal properties, and the corresponding flat cluster methods are classified and compared with known methods. The ten normal properties specify three distinct sequences of flat cluster methods based on the graph-theoretic concepts of k -minimum-degree, k -line-connectivity, and k -point-connectivity.