  We give a simpler proof using automata theory of a recent result of Kapovich,
Weidmann and Myasnikov according to which so-called benign graphs of groups
preserve decidability of the generalized word problem. These include graphs of
groups in which edge groups are polycyclic-by-finite and vertex groups are
either locally quasiconvex hyperbolic or polycyclic-by-finite and so in
particular chordal graph groups (right-angled Artin groups).
