Rank-non-increasing transformations on transition graphs

Abstract

This paper is concerned with cycle rank (or, as sometimes called, cycle complexity) of finite transition graphs, a notion which was introduced by Eggan in connection with the notion of star height of regular events. Certain basic types of transformations on transition graphs are introduced and their effect on the cycle rank of the graphs is investigated. It is then shown that any transition graph G can be converted, by a finite series of such transformations, into an equivalent non-deterministic state graph (i.e., transition graph without any empty-word transitions) which has no more nodes than G and no higher cycle rank. A stronger version of Eggan's Star Height Theorem follows.