1-String CZ-Representation of Planar Graphs

TL;DR

This paper proves that all planar 4-connected graphs can be represented using a specific string representation called CZ-representation, with each vertex represented by a path in a grid, ensuring a one-to-one correspondence between intersections and edges.

Contribution

It introduces a new CZ-representation for planar 4-connected graphs, demonstrating a grid size of n by 2n and a precise intersection property for edges.

Findings

Every planar 4-connected graph has a CZ-representation.

The representation uses paths with at most one vertical segment.

The grid size needed is n by 2n.

Abstract

In this paper, we prove that every planar 4-connected graph has a CZ-representation---a string representation using paths in a rectangular grid that contain at most one vertical segment. Furthermore, two paths representing vertices $u,v$ intersect precisely once whenever there is an edge between $u$ and $v$. The required size of the grid is $n \times 2n$.