Let $\Gamma\_{g,b}^m$ denote the mapping class group of a genus $g$ surface with $b$ non-permutable parametrised boundary curves and $m$ permutable punctures.


It is clear that in general, presentations for these groups are hard. I only need for the general case a set of generators (I want to show that two morphisms $\varphi,\psi:\Gamma\to G$ are equal and I want to check this on the generators). Partial answers are the following:


1. If $b,m=0$ (so we have a closed surface), then the group is generated by Dehn twists which can be easily drawn on the surface.
2. If $g,m=0$, we can declare one boundary curve as “outer” and the group is generated by Dehn twists along the inner boundary curves and the pure braid generators $\alpha\_{ij}$.


From Farb–Margalit, I know that there are always finitely many Dehn twists (or half twists) which generate $\Gamma$, but can we say in general where they are?



Generators of the mapping class group for surfaces with punctures and boundaries
See the paper


B. Wajnryb, "An elementary approach to the mapping class group of a surface," 
Geometry & Topology 3 (1999) 405–466.


See also "A finite presentation of the mapping class group of an oriented surface," by Gervais: <https://arxiv.org/pdf/math/9811162.pdf>.


