  Under investigation in this work is the robust inverse scattering transform
of the discrete Hirota equation with nonzero boundary conditions, which is
applied to solve simultaneously arbitrary-order poles on the branch points and
spectral singularities. Using the inverse scattering transform method, we
construct the Darboux transformation but not with the limit progress, which is
more convenient than before. Several kinds of rational solutions are derived in
detail. These solutions contain W-shape solitons, breathers, high-order rogue
waves, and various interactions between solitons and breathers. Moreover, we
analyze some remarkable characteristics of rational solutions through graphics.
Our results are useful to explain the related nonlinear wave phenomena.
