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We establish links between countable algebraically closed graphs and theendomorphisms of the countable universal graph $R$. As a consequence we showthat, for any countable graph $\Gamma$, there are uncountably many maximalsubgroups of the endomorphism monoid of $R$ isomorphic to the automorphismgroup of $\Gamma$. Further structural information about End $R$ is establishedincluding that Aut $\Gamma$ arises in uncountably many ways as aSch\"{u}tzenberger group. Similar results are proved for the countableuniversal directed graph and the countable universal bipartite graph.

Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph