A Method of Determining the Number of Vertices Contained in the SC Network Corresponding to a Given SC Transmission Function

Abstract

Directed graphs are often models for discrete sequential systems like computer programs, code generators, Markov processes, linear sampled-data systems, etc. [1]-[11]. One can study the behavior of such systems in a unified manner by the generating and characteristic functions [1], [2], [9]. We present here certain algorithms, suitable for digital computer mechanization, viz., 1) detection of certain structural ural flws (ill-formation) in the graph with respect to a set of initial and terminal nodes, 2) determination of redundant nodes, 3) enumeration and determination of all maximal strongly connected (M.S.C.) subgraphs, 4) determination of entries and exits of M.S.C. subgraphs, 5) partitioning of a graph into its component disjoint subgraphs. Whitney [11] and others [4]-[10] have considered the connectivity of graphs. Since we use the generating function, our results and methods in most cases are distinct and different. Also, our approach makes many physical applications meaningful. Since the generating function is based on a set of starting and terminating points, it provides a direct analog to computer programs and electrical circuits, which are characterized by entry and exit points, and sources and sinks, respectively. The concept of irredundant nodes is similar to that of [7], but we give explicit methods to determine the same. The method of enumerating the number of M.S.C. subgraphs and their identification is simpler than those of [7], [8], [10] from the digital computer mechanization viewpoint. Our methods of determination of entries and exits of M.S.C.