A combinatorial algorithm to compute presentations of mapping-class groups of orientable surfaces with one boundary component

We give an algorithm which computes a presentation for a subgroup, denoted $\AM_{g,1,p}$, of the automorphism group of a free group. It is known that $\AM_{g,1,p}$ is isomorphic to the mapping-class group of an orientable genus-$g$ surface with one boundary component and $p$ punctures. We define a variation of Auter space.