  We initiate the study of the vertex-ply of straight-line drawings, as a
relaxation of the recently introduced ply number. Consider the disks centered
at each vertex with radius equal to half the length of the longest edge
incident to the vertex. The vertex-ply of a drawing is determined by the vertex
covered by the maximum number of disks. The main motivation for considering
this relaxation is to relate the concept of ply to proximity drawings. In fact,
if we interpret the set of disks as proximity regions, a drawing with
vertex-ply number 1 can be seen as a weak proximity drawing, which we call
empty-ply drawing. We show non-trivial relationships between the ply number and
the vertex-ply number. Then, we focus on empty-ply drawings, proving some
properties and studying what classes of graphs admit such drawings. Finally, we
prove a lower bound on the ply and the vertex-ply of planar drawings.
