Computing roots of directed graphs is graph isomorphism hard

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The k-th power Dᵏ of a directed graph D is defined to be the directed graph on the vertices of D with an arc from a to b in Dᵏ iff one can get from a to b in D with exactly k steps. This notion is equivalent to the k-fold composition of binary relations or k-th powers of Boolean matrices. A k-th root of a directed graph D is another directed graph R with Rᵏ = D. We show that for each k >= 2, computing a k-th root of a directed graph is at least as hard as the graph isomorphism problem.