  The mapping class group of a genus $g$ surface $\Sigma_{g,1}$ with one
boundary component is known to have a simple yet infinite presentation with
generators given by elementary moves called Whitehead moves on so-called marked
bordered fatgraphs. In this paper, we introduce an algorithm called "fatgraph
Nielsen reduction" which, from the action of a mapping class $\varphi\in
MC_{g,1}$ of $\Sigma_{g,1}$ on the fundamental group $\pi_1(\Sigma_{g,1})$ of
$\Sigma_{g,1}$, determines a sequence of Whitehead moves representing $\varphi$
beginning at any choice of marked bordered fatgraph. As a consequence, this
leads to an algorithm which factors any mapping class given by its action on
$\pi(\Sigma_{g,1})$ in terms of a certain generating set for $MC_{g,1}$.
