There are n points on a graph. m pairs of these points are joined. No two intersect each other. Now (m+1)th pair has to drawn such that it doesn't intersect with any other line segment. Given number of points and pairs connected, is making (m+1)th line segment is impossible.
eg: 3 points and 2 pairs : 1-2 and 2-3. Now connecting a 3rd pair is not imposible. so this is accepted.
How to tell if in the given situation, connecting another pair is impossible or not?
Do the line segments have to be straight or can they curve? You can get an answer by looking at the Euler characteristic for convex polyhedra. If you graph contains a face with more than edges then you can draw an extra line. Try to draw graphs that only contain triangular faces, and guess a relation between V, E, and F. Subsequently try to prove it by induction
https://en.wikipedia.org/wiki/Euler_characteristic