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The main result of this paper is an algorithmic answer to the question raisedin the title, up to replacing the given $\hat{\phi} \in Out(F_n)$ by a positivepower.  In order to provide this algorithm, it is shown that every polynomiallygrowing automorphism $\hat \phi$ can be represented by an iterated Dehn twiston some graph-of-groups $\cal{G}$ with $\pi_1{\cal{G}} = F_n$. One then usesresults of two previous papers \cite{KY01, KY02} as well as some classicalresults such as the Whitehead algorithm to prove the claim.

When is a polynomially growing automorphism of $F_n$ geometric ?