We can see that a point $(x, y)$ is a vertex if and only if there are exactly one or three #’s in the four cells surrounding the point $(S_{x, y}, S_{x + 1, y}, S_{x, y + 1}, S_{x + 1, y + 1})$.

Note:
If there are exactly $2$ #’s around them, it is not possible that they are placed diagonally, since the shape does not have self-intersection. Therefore they are placed adjacently, thus the circumference of the polygon does not bend there, so it is not necessarily a vertex.

Since all the edge rows and columns consists of ., we only have to check for each $1 \le x \lt H, 1 \le y \lt W$, so we don’t really have to worry about referencing out of bounds of $S$.