Fixing Numbers of Graphs with Symmetric and Generalized Quaternion Symmetry Groups

Abstract

The fixing number of a graph \(\Gamma \) is the minimum number of vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of \(\Gamma \). This concept was extended to finite groups by Gibbons and Laison. The fixing set of a finite group G is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to G. Surprisingly few fixing sets of groups have been established; only the fixing sets of abelian groups and dihedral groups are completely understood. However, the fixing sets of symmetric groups have been studied previously. In this article, we establish new elements of the fixing sets of symmetric groups by considering line graphs of complete graphs. We conclude by establishing the fixing sets of generalized quaternion groups.