## Abstract

A countable graph is *ultrahomogeneous* if every isomorphism between
finite induced subgraphs can be extended to an automorphism. Woodrow and
Lachlan showed that there are essentially four types of such countably
infinite graphs: the random graph, infinite disjoint unions of complete
graphs ${K}_{n}$ with $n\in \mathbb{N}$ vertices, the ${K}_{n}$-free graphs, finite unions of the
infinite complete graph ${K}_{\omega}$, and duals of such graphs. The groups
$\mathrm{Aut}(\mathrm{\Gamma})$ of automorphisms of such graphs Γ have a natural
topology, which is compatible with multiplication and inversion, i.e. the
groups $\mathrm{Aut}(\mathrm{\Gamma})$ are topological groups. We consider the problem of
finding minimally generated dense subgroups of the groups $\mathrm{Aut}(\mathrm{\Gamma})$
where Γ is ultrahomogeneous. We show that if Γ is
ultrahomogeneous, then $\mathrm{Aut}(\mathrm{\Gamma})$ has 2-generated dense subgroups, and
that under certain conditions given $f\in \mathrm{Aut}(\mathrm{\Gamma})$ there exists $g\in \mathrm{Aut}(\mathrm{\Gamma})$ such that the subgroup generated by *f* and *g* is dense. We
also show that, roughly speaking, *g* can be chosen with a high degree of
freedom. For example, if Γ is either an infinite disjoint union of
${K}_{n}$ or a finite union of ${K}_{\omega}$, then *g* can be chosen to have any
given finite set of orbit representatives.

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