The [center of a graph][1] $G$ is the set of vertices that minimize the largest distance to vertices in $G$, e.g., in the graph below, that radius is $4$: <br /> ![GraphCenter50][2] <br /> 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 $C$? <hr /> (**Addendum** *5Mar14*.) Joe Malkevitch asked (personal communication): > Is every plane graph the center of some other plane graph? [1]: http://en.wikipedia.org/wiki/Graph_center [2]: https://i.sstatic.net/NcReI.jpg