F-Sets in graphs

Abstract

A subset S of the vertex set of a graph G is called an F-set if every α ϵ Γ( G), the automorphism group of G, is completely specified by specifying the images under α of all the points of S, and S has a minimum number of points. The number of points, k( G), in an F-set is an invariant of G, whose properties are studied in this paper. For a finite group Γ we define k( Γ) = max{ k( G) | Γ( G) = Γ}. Graphs with a given Abelian group and given k-value ( k ≤ k( Γ)) have been constructed. Graphs with a given group and k-value 1 are constructed which give simple proofs to the theorems of Frucht and Bouwer on the existence of graphs with given abstract/permutation groups.