Equilateral L-Contact Graphs

TL;DR

This paper explores contact graphs of axis-aligned L-shapes, providing characterizations, conversions to equilateral forms, and connections to planar graph representations, with implications for efficient algorithms.

Contribution

It introduces new characterizations of L-graphs, shows conversion to equilateral L-shapes, and links these to homothetic triangle contact representations, advancing understanding of planar graph contact systems.

Findings

Every contact system of L's can be converted to an equivalent equilateral L-system.

Equilateral L-contact systems are equivalent to homothetic triangle contact representations.

Every planar graph can be represented as a contact system of square-based cuboids.

Abstract

We consider {\em L-graphs}, that is contact graphs of axis-aligned L-shapes in the plane, all with the same rotation. We provide several characterizations of L-graphs, drawing connections to Schnyder realizers and canonical orders of maximally planar graphs. We show that every contact system of L's can always be converted to an equivalent one with equilateral L's. This can be used to show a stronger version of a result of Thomassen, namely, that every planar graph can be represented as a contact system of square-based cuboids. We also study a slightly more restricted version of equilateral L-contact systems and show that these are equivalent to homothetic triangle contact representations of maximally planar graphs. We believe that this new interpretation of the problem might allow for efficient algorithms to find homothetic triangle contact representations, that do not use Schramm's monster packing theorem.