creating a vertex without creating triangles?

Is it possible to create a vertex on an edge without cutting or subdivding the edge? as this usually creates unwanted triangles on my surfaces.

This raises the question as to how can I determine which vertices join to each other via lines and which don’t?

Currently with blender, no if you want it to be part of that edge. You can have a vertex that lies on the edge but is not connected to any other vertex.

Richard