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We completely classify redundant blow-ups appearing in the theory of rationalsurfaces with big anticanonical divisor due to Sakai. In particular, weconstruct a rational surface with big anticanonical divisor which is not aminimal 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.

Redundant blow-ups of rational surfaces with big anticanonical divisor