Planar graphs as L-intersection or L-contact graphs

TL;DR

This paper proves that all planar graphs can be represented as L-intersection graphs and that triangle-free planar graphs can be represented as contact graphs of L, |, and -- shapes, simplifying previous proofs.

Contribution

It establishes that planar graphs are L-intersection graphs and triangle-free planar graphs are {L, |, --}-contact graphs, confirming conjectures from 2013 with a new decomposition method.

Findings

Planar graphs are L-intersection graphs.

Triangle-free planar graphs are {L, |, --}-contact graphs.

Simplifies proof that planar graphs are segment intersection graphs.

Abstract

The L-intersection graphs are the graphs that have a representation as intersection graphs of axis parallel shapes in the plane. A subfamily of these graphs are {L, |, --}-contact graphs which are the contact graphs of axis parallel L, |, and -- shapes in the plane. We prove here two results that were conjectured by Chaplick and Ueckerdt in 2013. We show that planar graphs are L-intersection graphs, and that triangle-free planar graphs are {L, |, --}-contact graphs. These results are obtained by a new and simple decomposition technique for 4-connected triangulations. Our results also provide a much simpler proof of the known fact that planar graphs are segment intersection graphs.