Is there a word for "part of a route"?

I'm building a software application where I calculate which way to take between two different places. I call the complete path for a Route (correct?). But what do I call each part of the route?

The user wants to go from A to D, but to reach D the user must take the path A -> B -> C -> D.

What is part A->B called?

In computer science, I've heard this called a hop or a segment. In more general English usage, I think the term leg would be most appropriate. As for the whole sequence, route seems fine.

As Kristopher points out, edge is a technical term for what you're describing in graph theory, but an average user would have no idea what that meant. Only use this term if you're explicitly using the language of graph theory.

People usually call segments of a trip "legs":

The first leg of the trip will take us to Philadelphia. Then the second and third legs will take us to Dayton, OH and Chicago.

If the person using your program is deciding where each of the main stops are, each segment is called a "leg". Most routing software uses this terminology when planning multi-stop itineraries, assuming that the user is driving or otherwise in control of the vehicle. When applied to commercial flights, we generally use the term "leg" for each flight, as well, even though the customer is not necessarily the one making the choice about each stop.

However, we don't generally use the term "leg" to refer to every change in path, every navigation action. Better terms would be a "turn", "step", or, possibly, "waypoint." ("Waypoint" has some of the connotations of choice that "leg" has, however). Most users won't think of the "edge" as the important thing, but if they did, you could call that a "route segment." For most users of turn-by-turn navigation systems, the vertex (turn) is the thing that they focus on, rather than the edge (segment).

You should take a look at DDD Sample. Its domain model is similar to yours and, in particular, the whole way is called itinerary and each segment is called leg. But the act of calculating possible itineraries is called routing.