  We prove that the redundant blow-up preserves the finite generation of the
Cox ring of a rational surface under a suitable assumption, and we study the
birational structure of Mori dream rational surfaces via redundant blow-ups. It
turns out that the redundant blow-up completely characterizes birational
morphisms of Mori dream rational surfaces with anticanonical Iitaka dimension
$0$. As an application, we construct new Mori dream rational surfaces with
anticanonical Iitaka dimension $0$ and $-\infty$ of arbitrarily large Picard
number.
