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

We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $\mathrm{\Gamma }$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $\mathrm{\Gamma }$. Further structural information about End $R$ is established including that Aut $\mathrm{\Gamma }$ arises in uncountably many ways as a Sch"{u}tzenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.