The main result of this paper is an algorithmic answer to the question raised
in the title, up to replacing the given ϕ^​∈Out(Fn​) by a positive
power.
In order to provide this algorithm, it is shown that every polynomially
growing automorphism ϕ^​ can be represented by an iterated Dehn twist
on some graph-of-groups G with π1​G=Fn​. One then uses
results of two previous papers \cite{KY01, KY02} as well as some classical
results such as the Whitehead algorithm to prove the claim