We give a sufficient condition under which the fundamental group of a reglued
graph of surfaces is hyperbolic. A reglued graph of surfaces is constructed by
cutting a fixed graph of surfaces along the edge surfaces, then regluing by
pseudo-Anosov homeomorphisms of the edge surfaces. By carefully choosing the
regluing homeomorphism, we construct an example of such a reglued graph of
surfaces, whose fundamental group is not abstractly commensurate to any
surface-by-free group, i.e., which is different from all the examples given in
Mosher's paper 'A hyperbolic-by-hyperbolic hyperbolic group'.