Let $G$ be a directed graph without loop and suppose that $M$ is its endomorphism monoid.
First how we can create a simple graph $\Gamma$ (using $G$), with the same endomorphism monoid? and then how we can create it with the smallest possible number of vertices?


Thanks for your help 



The smallest number of vertices for a graph with the same endomorphism monoid
These kinds of results are considered in the paper <http://onlinelibrary.wiley.com/doi/10.1002/jgt.20396/epdf>


