Convex Grid Drawings of Planar Graphs with Constant Edge-Vertex Resolution

TL;DR

This paper introduces a new algorithm for drawing 3-connected planar graphs convexly on a grid, ensuring a constant minimum distance between edges and vertices, with improved area bounds over previous methods.

Contribution

The paper presents a novel convex grid drawing algorithm for 3-connected plane graphs that guarantees a constant edge-vertex resolution and improves area bounds.

Findings

Achieves edge-vertex resolution at least 1/2

Uses an integer grid of size (n-2+a) x (n-2+a)

Improves previous area bounds for convex drawings

Abstract

We continue the study of the area requirement of convex straight-line grid drawings of 3-connected plane graphs, which has been intensively investigated in the last decades. Motivated by applications, such as graph editors, we additionally require the obtained drawings to have bounded edge-vertex resolution, that is, the closest distance between a vertex and any non-incident edge is lower bounded by a constant that does not depend on the size of the graph. We present a drawing algorithm that takes as input a 3-connected plane graph with n vertices and f internal faces and computes a convex straight-line drawing with edge-vertex resolution at least 1/2 on an integer grid of size (n-2+a)x(n-2+a), where a=min{n-3,f}. Our result improves the previously best-known area bound of (3n-7)x(3n-7)/2 by Chrobak, Goodrich and Tamassia.