Moving only one vertex of the edge

Question

I searched for this, however I'm probably not familiar with the terminology, so could not find what I wanted. Is there a way to move only one vertex of the edge? Example:

I've added a cube, then subdivided the top and bottom edges and added two edges from the top-left and top-middle vertices to the bottom-right vertex. I want to move only the "left" vertex (top-left to bottom-right - selected on the left-most cube) to the bottom-middle vertex, so I arrive at either of the two right-most configurations below.

I understand I can delete the edge and create another one (which is what I did to demonstrate the above anyway), just wondering if this can be done to save time when moving several vertices at once.

I guess the main reason is to be able to manually refine things such as using Alt+F to join a cylinder and a cube, getting this:

So it would be easier to move the three selected ones to the bottom-middle vertex, instead of deleting them and then creating the three new ones.

Of course, regardless of the above question, there might be an easier way to do what I showed in the lase image - open to suggestions here. The point is to join two faces together so that the join is smooth.

Answer 1

You can use AltV for Rip fill.

Normally you could just press G with the vertex selected (or GG if you wanted to slide a vertex along the edge).
However, since the vertex you want to move has other edges connected to it, you must create a new, separate vertex with the one edge attached (Rip, V), and then fill in the newly formed hole (Fill, F). The Rip Fill tool does these two operations automatically.

I assumed that there is only one vertex at the corner. If you have multiple vertices in the same place (doubles), then you need only to press G with the right one selected.