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×2n.
@article{arxiv.1409.5816,
title = {1-String CZ-Representation of Planar Graphs},
author = {Therese Biedl and Martin Derka},
journal= {arXiv preprint arXiv:1409.5816},
year = {2014}
}