  A solution of the isomorphism problem is presented for the class of Coxeter
groups W that have a finite set of Coxeter generators S such that the
underlying graph of the presentation diagram of the system (W,S) has the
property that every cycle of length at least four has a cord. As an
application, we construct counterexamples to two main conjectures concerning
the isomorphism problem for Coxeter groups.
