Low ply graph drawing

We consider the problem of characterizing graphs with low ply number and algorithms for creating layouts of graphs with low ply number. Informally, the ply number of a straight-line drawing of a graph is defined as the maximum number of overlapping disks, where each disk is associated with a vertex and has a radius that is half the length of the longest edge incident to that vertex. We show that internally triangulated biconnected planar graphs that admit a drawing with ply number 1 can be recognized in O(n log n) time, while the problem is in general NP-hard. We also show several classes of graphs that have 1-ply drawings. We then show that binary trees, stars, and caterpillars have 2-ply drawings, while general trees have (h+1)-ply drawings, where h is the height of the tree. Finally we discuss some generalizations of the notion of a ply number.