You can graph them as two separate line segments. Imagine that you have a graph with the horizontal axis being distance and the vertical axis being time. The departure point is (0,0), so both timelines start there. The point of return is (0,T) where T is the time that the at-home twin waited for the traveller's return. Now you can draw the at-home twin's timeline as a vertical line from (0,0) to (0,T), and the travelers timeline sloping out to (D,T/2) where D is the distance travelled, and then another line sloping from there back to (0,T).
(If you already know this, I apologize for restating the elementary; you've used just enough non-standard terminology that I'm not sure if you're already familiar with these diagrams).
This is all very well, except that now you'll be looking at this graph and thinking that the traveller's timeline is longer, not shorter - it's two sides of a triangle. The trick is that in space-time the distance between two points is calculated as the square root of t^2-x^2, not the square root of t^2+x^2 as you expect from the ordinary Pythagorean theorem.
(Disclaimer: I've played fast and loose with several conventions here, bashed over some subtleties, and been completely sloppy about who is measuring D and T. You'll also have noticed that you can easily find the square root of a negative number popping up in some distance calculations... Well, whaddaya expect for just two paragraphs of answer?)