This question relates to [Realizing groups as automorphism groups of graphs.](https://mathoverflow.net/questions/37356/realizing-groups-as-automorphism-groups-of-graphs)


Given a [monoid](http://en.wikipedia.org/wiki/Monoid) $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?



Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$
Hedrlín and Pultr proved that every monoid was the endomorphism monoid of a graph. See their paper "Symmetric relations (undirected graphs) with given semigroup" Monatsh. Math 69 (1965), [eudml](https://eudml.org/doc/177296), DOI: [10.1007/BF01297617](https://doi.org/10.1007/BF01297617).


