TITLE: Is every graph the center of some other graph?
QUESTION [14 upvotes]: The center of a graph $G$ is the set of vertices that minimize the largest
distance to vertices in $G$, e.g., in the graph below, that radius is $4$:

 
 
 
 
 


Define the center $C$ as the subgraph of $G$ induced
by those vertices.
I seek to learn constraints on $C$. Is it the case that every graph $C$ is the center of some graph $G$? Or are there constraints on the possible structures of $C$?

(Addendum 5Mar14.)
Joe Malkevitch asked (personal communication):

Is every plane graph the center of some other plane graph?

REPLY [22 votes]: I'm just copying my comment above, as it seems to answer the question.  For any graph $C$ there exists a graph for which this is the center.  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 $A$, and this number is larger than the maximum length between two vertices in $C$).  Now the vertices in $C$ form the center of this new graph.