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Milligram
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Name the class of graphs G s.t. every two graphs that can be created by removing one edge from G are isomorphic.
Sleepless, I aimed at your first choice. Namely, the property is that $G_1$ is isomorphic to $G_2$ whenever $G_1$ is obtained by removing one edge from $G$ and $G_2$ is obtained by removing one edge from $G$.
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