[Asymmetric graphs](http://en.wikipedia.org/wiki/Asymmetric_graph) are graphs that have a trivial automorphism group $\textrm{Aut}(G)$, i.e. the only graph isomorphism from $G$ to itself is the identity.


Let's call a graph $G$ strongly asymmetric if the only graph *homomorphism* $h: G\to G$ is the identity (in other words, the endomorphism monoid $\textrm{End}(G)$ is trivial).


Given a (finite or infinite) cardinal $\kappa > 0$, is there a strongly asymmetric graph on $\kappa$ vertices?



Strongly asymmetric graphs
I believe the common name for such graphs is *rigid*. In fact, most random graphs are rigid. See this reference: [On the minimal order of a graphs within a semigroup](http://www.sciencedirect.com/science/article/pii/0095895684900212).


