  The anti-forcing number of a connected graph $G$ is the smallest number of
edges such that the remaining graph obtained by deleting these edges has a
unique perfect matching. In this paper, we show that the anti-forcing number of
every fullerene has at least four. We give a procedure to construct all
fullerenes whose anti-forcing numbers achieve the lower bound four.
Furthermore, we show that, for every even $n\geq20$ ($n\neq22,26$), there
exists a fullerene with $n$ vertices that has the anti-forcing number four, and
the fullerene with 26 vertices has the anti-forcing number five.
