Abstract
The ⌞-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 {⌞, |, −}-contact graphs which are the contact graphs of axis parallel ⌞, |, 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 ⌞-intersection graphs, and that triangle-free planar graphs are {⌞, |, −}-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.
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