  We show that any free product of two countable groups, one of them being
infinite, admits a faithful and homogeneous action on the Random Graph. We also
show that a large class of HNN extensions or free products, amalgamated over a
finite group, admit such an action and we extend our results to groups acting
on trees. Finally, we show the ubiquity of finitely generated free dense
subgroups of the automorphism group of the Random Graph whose action on it have
all orbits infinite.
