Segment representations with small resolution

TL;DR

This paper investigates the minimal grid resolution needed for segment representations of graphs, proving that all planar graphs can be represented within a grid of size exponential in the number of vertices.

Contribution

It establishes an upper bound on the grid resolution for segment representations of planar graphs and related classes, demonstrating that such representations can be achieved within exponential grid size.

Findings

All planar graphs have segment representations on a grid of size 4^n by 4^n.

The result applies to graphs with L-representations.

Provides bounds on grid resolution for segment graph representations.

Abstract

A segment representation of a graph is an assignment of line segments in 2D to the vertices in such a way that two segments intersect if and only if the corresponding vertices are adjacent. Not all graphs have such segment representations, but they exist, for example, for all planar graphs. In this note, we study the resolution that can be achieved for segment representations, presuming the ends of segments must be on integer grid points. We show that any planar graph (and more generally, any graph that has a so-called $L$-representation) has a segment representation in a grid of width and height $4^n$.