Divide overlapping objects with strokes

I have a bunch of overlapping objects with strokes and no fills. I want to divide them in the same way that I divide filled objects.

What I want is a cut where the overlapping paths intersect, so I can delete the unwanted path segments.

I know that it's possible to use the Knife or Eraser to do this "by hand", but what I'm looking for is a one-click procedure.

Is there a way to do this?