Groups, Graphs, and Symmetry-Breaking
Karen Sue Potanka · 1998
A labeling of a graph G is said to be r-distinguishing if no automorphism of G preserves all of the vertex labels. The smallest such number r for which there is an r-distinguishing labeling on G is called the distinguishing number of G. Thedistinguishing set of a group Γ, D(Γ), is the set of distinguishing numbers of graphs G in which Aut(G) ∼ = Γ. It is shown that D(Γ) is non-empty for any finite group Γ. In particular, D(Dn) is found where Dn is the dihedral group with 2n elements. From there, the generalized Petersen graphs, GP (n, k), are defined and the automorphism groups and distinguishing numbers of such graphs are given.