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{\bf Stephen G. Hartke, Hannah Kolb, Jared Nishikawa and Derrick Stolee}
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{\bf Automorphism Groups of a Graph and a Vertex-Deleted Subgraph}
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Understanding the structure of a graph along with the structure of its
subgraphs is important for several problems in graph theory. Two
examples are the Reconstruction Conjecture and isomorph-free
generation. This paper raises the question of which pairs of groups
can be represented as the automorphism groups of a graph and a
vertex-deleted subgraph.  This, and more surprisingly the analogous
question for edge-deleted subgraphs, are answered in the most positive sense using concrete constructions.

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