  In this paper, we identify two fractals if and only if they are biLipschitz
equivalent. Fix ratio $r,$ for dust-like graph-directed sets with ratio $r$ and
integer characteristics, we show that they are rigid in the sense that they are
uniquely determined by their Hausdorff dimensions. Using this rigidity theorem,
we show that in some class of self-similar sets, two totally disconnected
self-similar sets without complete overlaps are biLipschitz equivalent.
