  We introduce a novel combinatorial method to study $Q^{**}$-transformations
of group presentations or, equivalently, 3-deformations of CW-complexes of
dimension 2. Our procedure is based on a refinement of discrete Morse theory
that gives a Whitehead simple homotopy equivalence from a regular CW-complex to
the simplified Morse CW-complex, with an explicit description of the attaching
maps and bounds on the dimension of the complexes involved in the deformation.
We apply this technique to show that some known potential counterexamples to
the Andrews--Curtis conjecture do satisfy the conjecture.
