Redundant blow-ups of rational surfaces with big anticanonical divisor

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.