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 Gamma, there are uncountably many maximal subgroups of the endomorphism monoid of R isomorphic to the automorphism group of Gamma. Further structural information about End R is established including that Aut Gamma arises in uncountably many ways as a Schutzenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.