Is every graph the center of some other graph? @QiaochuYuan: I thought it would take one more step to connect $A$ to the end of $B$, and so $A$ would not be in the center; and the same for $B$. Am I missing something?

Mar 2

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Is every graph the center of some other graph? How about the following construction: Introduce two new vertices $A$ and $B$. Connect all the vertices in $C$ to $A$; and connect all the vertices in $C$ to $B$. From $A$ draw a long line with many vertices on it (the line doesn't have any other edges to $C$), and similarly from $B$ draw a long line with many vertices on it (same number of vertices as the line from $A$). Now if I understand your question correctly, I think $C$ is the center of this new graph.