  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.
