  The computational complexity of the word problem in HNN-extension of groups
is studied. HNN-extension is a fundamental construction in combinatorial group
theory. It is shown that the word problem for an ascending HNN-extension of a
group H is logspace reducible to the so-called compressed word problem for H.
The main result of the paper states that the word problem for an HNN-extension
of a hyperbolic group H with cyclic associated subgroups can be solved in
polynomial time. This result can be easily extended to fundamental groups of
graphs of groups with hyperbolic vertex groups and cyclic edge groups.
