  We define strict confluent drawing, a form of confluent drawing in which the
existence of an edge is indicated by the presence of a smooth path through a
system of arcs and junctions (without crossings), and in which such a path, if
it exists, must be unique. We prove that it is NP-complete to determine whether
a given graph has a strict confluent drawing but polynomial to determine
whether it has an outerplanar strict confluent drawing with a fixed vertex
ordering (a drawing within a disk, with the vertices placed in a given order on
the boundary).
