  We propose a simple, exactly solvable, model of interface growth in a random
medium that is a variant of the zero-temperature random-field Ising model on
the Cayley tree. This model is shown to have a phase diagram (critical
depinning field versus disorder strength) qualitatively similar to that
obtained numerically on the cubic lattice. We then introduce a specifically
tailored random graph that allows an exact asymptotic analysis of the height
and width of the interface. We characterize the change of morphology of the
interface as a function of the disorder strength, a change that is found to
take place at a multicritical point along the depinning-transition line.
