Bold Graph Drawings

When a graph is drawn in a classical manner, its vertices are shown as small disks and its edges with a positive width; zero-width edges exist only in theory. Let r denote the radius of the disks that show vertices and w the width of edges. We give a list of conditions that make such a drawing good and that apply to not necessarily planar graphs. We show that if r < w, a vertex must have constant degree for a drawing to satisfy the conditions, and if r ≥ w, a vertex can have any degree. We also give an algorithm that, for a given drawing and a ratio like r = 2w, computes the maximum r and w without violating the conditions.