Isolated Knife cut

Does anyone know if there is a way to isolate a knife cut within certain faces or vertices?

If I take a mesh like this…

And then I make a cut on the selected face like so…

As you can see, now we have all of these extra unwanted vertices. Is there any way to not have this happen or remove these??

That’s because knife is edgebased, and you can’t simply cut from vert to vert. A solution like proposed here (Look for Cut threshold):
would solve this.