Reference to extended Frucht's theorem: $\operatorname{Aut}(\Gamma)=G_1$ and $\operatorname{Aut}(\Gamma-e)=G_2$

Question

I remember hearing an extension of Frucht's theorem that goes something like the following:

Let $G_1$ and $G_2$ be finite groups. Then there exists a (finite) graph $\Gamma$ such that $\operatorname{Aut}(\Gamma)=G_1$ and $\operatorname{Aut}(\Gamma-e)=G_2$ for some edge $e\in E(\Gamma)$.

Can someone point me to a reference to this theorem? (Or, if I am remembering it incorrectly, is there a similar theorem I might be mixing it up with?)

Answer 1

See Theorem 4.2 in S. G. Hartke, H. Kolb, J. Nishikawa, and D. Stolee, Automorphism groups of a graph and a vertex-deleted subgraph. Elec. J. Combin., 17 (2010), # R134, 8pp. PDF link.

Theorem 4.2. If $\Gamma_1$ and $\Gamma_2$ are groups, then there exists a graph $G$ and an edge $e\in E(G)$ so that $\Gamma_1\stackrel{G-e}\longrightarrow\Gamma_2$.

Where $\Gamma_1\stackrel{G-e}\longrightarrow\Gamma_2$ is defined in Definition 2.1 as follows:

Definition 2.1. ...If a specific graph $G$ and subobject $x$ give Aut$(G)\simeq\Gamma_1$ and Aut$(G-x)\simeq\Gamma_2$, the deletion relation may be presented as $\Gamma_1\stackrel{G-x}\longrightarrow\Gamma_2$.