This question relates to http://mathoverflow.net/questions/37356/realizing-groups-as-automorphism-groups-of-graphs

Given a [monoid][1] $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?


  [1]: http://en.wikipedia.org/wiki/Monoid