Simple Morse functions on an oriented surface with boundary

In the paper, smooth functions with non-degenerate critical points on a smooth compact surface with boundary are considered. Firstly, it is shown that these functions are topologically equivalent to $m$-functions. The equipped Reeb graph is used to describe their topological structure. Secondly, the authors characterize the topological structure of all simple functions with at most 5 critical points. And finally, a formula for the genus of the surface based on the equipped Reeb graph is obtained.