  In this article we construct asynchronous and sometimes synchronous automatic
structures for amalgamated products and HNN extensions of groups that are
strongly asynchronously (or synchronously) coset automatic with respect to the
associated automatic subgroups, subject to further geometric conditions. These
results are proved in the general context of fundamental groups of graphs of
groups. The hypotheses of our closure results are satisfied in a variety of
examples such as Artin groups of sufficiently large type, Coxeter groups,
virtually abelian groups, and groups that are hyperbolic relative to virtually
abelian subgroups.
