On Vertex- and Empty-Ply Proximity Drawings

TL;DR

This paper introduces the concept of vertex-ply in straight-line graph drawings, explores its relationship with ply number, and investigates properties and limitations of empty-ply (weak proximity) drawings, especially for planar graphs.

Contribution

It defines vertex-ply as a relaxation of ply number, analyzes its properties, and studies the existence of empty-ply drawings across different graph classes, including planar graphs.

Findings

Non-trivial relationships between ply number and vertex-ply.

Properties of empty-ply drawings established.

Lower bounds on ply and vertex-ply for planar drawings proved.

Abstract

We initiate the study of the vertex-ply of straight-line drawings, as a relaxation of the recently introduced ply number. Consider the disks centered at each vertex with radius equal to half the length of the longest edge incident to the vertex. The vertex-ply of a drawing is determined by the vertex covered by the maximum number of disks. The main motivation for considering this relaxation is to relate the concept of ply to proximity drawings. In fact, if we interpret the set of disks as proximity regions, a drawing with vertex-ply number 1 can be seen as a weak proximity drawing, which we call empty-ply drawing. We show non-trivial relationships between the ply number and the vertex-ply number. Then, we focus on empty-ply drawings, proving some properties and studying what classes of graphs admit such drawings. Finally, we prove a lower bound on the ply and the vertex-ply of planar drawings.