  We completely classify redundant blow-ups appearing in the theory of rational
surfaces with big anticanonical divisor due to Sakai. In particular, we
construct a rational surface with big anticanonical divisor which is not a
minimal resolution of a del Pezzo surface with only rational singularities,
which gives a negative answer to a question raised in a paper by Testa,
V\'{a}rilly-Alvarado, and Velasco.
