Can I select points without lines?

I have a design with many boxes that have points and edges intersecting, one on top of the other. I want to be able to box-select points and move them without selecting any lines.  If I try to click-select, I can't select two points that are stacked one on top of the other, and if I box-select, I get the edges/lines and can't move the points without moving the edges as well.  Is there any modifier key or anything to disable selecting of lines and be able to only select points? Here are images showing the two points selected to move, and what happens if they're moved up. The lines come along. It seems there must be a way. Thank you!