Rectangle-of-influence triangulations

The concept of rectangle-of-influence (RI) drawings is old and well-studied in the area of graph drawing. A graph has such a drawing if we can assign points to its vertices such that for every edge (v, w) the supporting rectangle (i.e., the minimal closed axis-aligned rectangle R(v, w) containing v and w) contains no other points. In the original setup, the graph had to have an edge for every pair of points with an empty supporting rectangle (the strong model, see e.g. [8]). Later papers focus on weak RI-drawings, where for every edge the supporting rectangle must be empty, but not all such edges must exist. Of particular interest are planar weak RI-drawings of planar graphs, since these can always be deformed to reside on an n× n-integer grid. See e.g. [9, 1].