Perfect Mesh Cutting Problem

Hi, I am currently working on a system that can create a cut anywhere on a dynamic 2D mesh (that I have generated from a script) depending on where the player clicks on the mesh with his/her mouse. (If you need an example I can easily provide it!)

I am having quite a bit of trouble with this, and I've be working on a solution for about a month now with no luck.

I searched around for a while and found this:
http://answers.unity3d.com/questions/379157/cutting-simple-plane-with-a-line-2d.html

Which is almost exactly what I am trying to do, but I found the answer too vague for me to figure out how to implement it.

Any help going into more detail with the answer shown above, or if you can suggest/give a different solution, would be greatly appreciated!

Thanks!!