Subdirect decompositions of transformation graphs

Abstract

A transformation graph is a finite directed graph with exactly one edge issuing from each vertex. A graph is subdirectly decomposed into two factor graphs, if it is isomorphic to a subgraph of their direct (Cartesian) product. The decomposition is A -optimal, if the maximal order (i.e., number of vertices) of the factor graphs is minimal. It is B -optimal, if the sum of the orders is minimal. This paper describes techniques for obtaining optimal decompositions of a given connected transformation graph. To achieve this objective, computationally convenient expressions are introduced for the isomorphism classes of transition graphs, i.e., finite, directed graphs with at most one edge issuing from each vertex. Then, formulas are derived for computing the expression of the direct (Cartesian) product of given transition graphs. The decomposition techniques are based on these expressions and formulas. The results obtained are directly applicable to the synthesis of autonomous sequential networks by means of parallel-connected smaller networks.