Detecting if a line whether a line runs from one vertex to another completely within a polygon

Question

I'm working with a case in path finding in java, in which I need to detect, whether a line intersects a polygon, for which I test, if the line intersect any of the polygons edges; but because I need to hop from vertex to vertex, I need to allow the corners of the polygon. Unfortunately this also allows an edge from a vertex to another that lies completely inside the polygon.

Example:

I'm looking for a way to detect this case. Does anyone have an idea?

Answer 1

In the case when diagonal does not intersect edges, additionally check whether middle of this diagonal lies inside polygon
(for example, using famous ray-edges crossing method)