What is this, a crossover episode

You are given a simple polygon (that is, its edges don't intersect with each other, apart from touching at the vertices) with $n$ vertices. The vertices are numbered, in clockwise order, as $1, 2, \ldots, n$. Additionally, there are $m$ non-intersecting chords within the polygon. Note however that chords can touch each other at a vertex of the polygon. A chord is defined to be a line segment conne