Abstract
We consider 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 max- imally 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.