TITLE: Generators of the mapping class group for surfaces with punctures and boundaries
QUESTION [5 upvotes]: 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:

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.
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?

REPLY [5 votes]: 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.