  A series of pairs of graphs (Gamma_k, Gamma'_k), k = 0,1,2,... has been
considered as candidates for dual pairs of principal graphs of subfactors of
small Jones index above 4 and it has recently been proved that the pair
(Gamma_k, Gamma'_k) comes from a subfactor if and only if k = 0 or k =1. We
show that nevertheless there exists a unique fusion system compatible with this
pair of graphs for all non-negative integers k.
