On a Shannon cover of certain reducible shift of finite type

A Shannon cover is a canonical presentation of a sofic shift and it is widely used to analyze the properties of a sofic shift. For the case when a sofic shift is irreducible, it is well known that a Shannon cover of the shift turns out to be unique (up to labeled graph isomorphism). Furthermore, there exists a necessary and sufficient condition for a given deterministic presentation to be the Shannon cover, and also, there is a procedure to obtain the Shannon cover from a given deterministic presentation. However, when a sofic shift is not irreducible (i.e., reducible), there can be two or more Shannon covers, and there is not known any specific condition for a presentation to be a Shannon cover. In this paper, we focus on a certain class of reducible shifts of finite type (SFT's), and show that a presentation obtained by following an algorithm introduced by Chrochemore, Mignosi and Restivo is a Shannon cover of an SFT in the class, together with the specific number of vertices in the presentation. Furthermore, we prove the uniqueness of a Shannon cover of such an SFT.