Tallinn. Translated from Avtomatika and Telemekhanika, No. 5, pp.130-139, May 1976, Original Article submitted April 29, 1975; No. 8, pp.169-178, August 1976, Original Article submitted August 13, 1975; No. 1, pp.109-119, January 1977, Original Article submitted February 23, 1976.
UDC 62-50:519.2

J.E. Mullat

Extremal Subsystems of Monotonic Systems, I (pdf)

A general theoretical method is described which is intended for the initial analysis of systems of interrelated elements. Within the framework of the model a specially postulated monotonicity property for systems guarantees the existence of a special kind of subsystems called kernels. A number of extremal properties and the structure of the kernels are found. The language of description of Monotonic Systems of interrelated elements is described in general set-theoretic terms and leads to a constructive system of notions in case of system with finite number of elements. A series of properties of special finite sequences of elements are studied whereby kernels in Monotonic System are classified.

Extremal Subsystems of Monotonic Systems, II (pdf)

A constructive procedure is considered for obtaining singular determining sequence of elements of Monotonic systems studied in I. The relationship between two determining sequences   a+   and   a-   is also examined, and the obtained result is formulated as a duality theorem. This theorem is used for describing a procedure of restricting the domain of search for extremal subsystems (or kernels of a Monotonic System): the corresponding search scheme is also presented.

Extremal Subsystems of Monotonic Systems, III (pdf)

An attempt is made to find parts of a given graph that are more "saturated" than any other part with "small" graphs of the same type. Based on such formulation, this problem is solved by constructing a monotonic system from structural elements of graphs (arcs or vertices). The scheme of producing a monotonic system from a given graph is presented in general form, and the necessary constructions are illustrated by examples. This paper is a continuation of I and II ; it has the purpose of illustrating the procedures (developed in the first two parts) of finding extremal subsystems for solving certain problems arising in tournaments, a-cyclic graphs, and undirected and directed trees.