Low Ply Drawings of Trees and 2-Trees

Ply number is a recently developed graph drawing metric inspired by studying road networks. Informally, for each vertex v, which is associated with a point in the plane, a disk is drawn centered on v with a radius that is alpha times the length of the longest edge incident to v, for some constant alpha in (0, 0.5]. The ply number is the maximum number of disks that overlap at a single point. We show that any tree with maximum degree Delta has a 1-ply drawing when alpha = O(1 / Delta). We also show that when alpha = 1/2, trees can be drawn with logarithmic ply number, with an area that is polynomial for bounded-degree trees. Lastly, we show that this logarithmic upper bound does not apply to 2-trees, by giving a lower bound of Omega(sqrt(n / log n)) ply for any value of alpha.