  For a subset $ S $ of $ \mathbb R^d$, $ S$-graphs are the intersection graphs
of specific transformations of $ S $. The class of Burling graphs is a class of
triangle-free graphs with arbitrarily large chromatic number that has attracted
much attention in the last years. In 2012, Pawlik, Kozik, Krawczyk, Laso\'n,
Micek, Trotter, and Walczak showed that for every compact and path-connected
set $ S \subseteq \mathbb R^2$ that is different from an axis-parallel
rectangle, the class of $ S $-graphs contains all Burling graphs. There is,
however, a gap between the two classes. In recent years, there have been
improvements in understanding the subclasses of $ S $-graphs that are closer or
equal to Burling graphs. In this article, we close this gap for every set $ S $
with the mentioned properties: we introduce the class of constrained $ S
$-graphs, a subclass of $ S$-graphs, and prove that it is equal to the class of
Burling graphs. We also introduce the class of constrained graphs, a subclass
of intersection graphs of subsets of $ \mathbb R^2$, and prove that it is equal
to the class of Burling graphs.
