  In this paper we investigate the integrability of two-dimensional partial
difference equations using the newly developed techniques of study of the
degree of the iterates. We show that while for generic, nonintegrable
equations, the degree grows exponentially fast, for integrable lattice
equations the degree growth is polynomial. The growth criterion is used in
order to obtain the integrable deautonomisations of the equations examined. In
the case of linearisable lattice equations we show that the degree growth is
slower than in the case of equations integrable through Inverse Scattering
Transform techniques.
