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Let $(K_1,\ldots,K_n)$ be a Cantor type graph-directed attractors in$\mathbb{R}^d$. By creating an auxiliary graph, we provide an effectivecriterion of whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leqn$. Moreover, the emptiness can be checked by examining only a finite number ofthe attractor's geometric approximations. The method is also applicable formore general graph-directed systems. As an application, we are able todetermine the connectedness of all $d$-dimensional generalized Sierpi\'nskisponges of which the corresponding IFSs are allowed to have rotational andreflectional components.

An intersection problem of graph-directed attractors and an application