Geodesic
Characterizations of Schützenberger graphs in terms of their automorphism groups and fundamental groups
Glasgow mathematical journal, Tome 35 (1993) no. 3, pp. 275-291

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The importance of the fundamental group of a graph in group theory has been well known for many years. The recent work of Meakin, Margolis and Stephen has shown how effective graph theoretic techniques can be in the study of word problems in inverse semigroups. Our goal here is to characterize those deterministic inverse word graphs that are Schutzenberger graphs and consider how deterministic inverse word graphs and Schutzenberger graphs can be constructed from subgroups of free groups. 
Cowan, David; Reilly, Norman R. Characterizations of Schützenberger graphs in terms of their automorphism groups and fundamental groups. Glasgow mathematical journal, Tome 35 (1993) no. 3, pp. 275-291. doi: 10.1017/S0017089500009861
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