Fewest traversals to visit all vertices of DAG

Question

I want to find the fewest traversals to visit all vertices of a DAG. To take a very simple case:

Boston -> New York   -> San Francisco or Las Vegas, both to -> LA
       -> Pittsburgh -> San Francisco or Las Vegas, both to -> LA

There are lots of ways to traverse this graph, but to visit every vertex, you only need:

1. Boston -> New York   -> San Francisco -> LA
2. Boston -> Pittsburgh -> Las Vegas     -> LA

I have no preferences in how the graph is visited, and it's fine to visit the same vertex multiple times during different traversals.

(Real application: I am generating test cases for human testers and want them to help them perform their tests at every 'vertex' with as few run-throughs as possible.)