  Nielsen reduction is an algorithm which decomposes any automorphism of a free
group into a product of elementary Nielsen transformations. While this may be
applied to a mapping class of a surface $S_{g,1}$ with one boundary component,
the resulting decomposition in general will not have a topological
interpretation. In this survey, we discuss a variation called fatgraph Nielsen
reduction which decomposes such a mapping class into elementary Nielsen
transformations interpreted as rearrangements of polygon domains for $S_{g,1}$
described by systems of arcs in $S_{g,1}$. These elementary moves generate the
chord slide groupoid of $S_{g,1}$, which we survey and describe in terms of
generators and relations.
