Smallest primitive embeddings of planar graphs

A graph is said to be planar if it can be drawn on the plane with vertices as different points and edges as continuous curves that only intersect its vertices. An embedding of a graph is said to be primitive if its edges are primitive segments. A recent conjecture is that all planar graphs with n vertices have a primitive embedding in a square grid of side O(n). It is known that trees have that type of embedding. A smallest primitive embedding is that in which the square grid has side as small as possible. In this work we present some results about the smallest primitive embeddings for trees, outerplanar graphs, and planar graphs with few vertices, as a computational approach to give evidence that the above conjecture might be true.