Star Height of Certain Families of Regular Events

This paper studies the relationship between the apparent star height of a given regular expression and the structure of its reduced deterministic state graph. Sufficient conditions for the star height of a regular event R to equal the cycle rank of its reduced state graph G"R are derived. The cycle rank of G"R is also shown to constitute a lower bound to the star height of certain subsets of R. These results are then applied to fully characterize the star height of events consisting of @? sets of paths in finite digraphs and two open problems posed by Eggan are answered.