  It was noted already in the 90s that many classic graph classes, such as
interval, chordal, and bipartite graphs, can be characterized by the existence
of an ordering of the vertices avoiding some ordered subgraphs, called
patterns. Very recently, all the classes corresponding to patterns on three
vertices (including the ones mentioned above) have been listed, and proved to
be efficiently recognizable. In contrast, very little is known about patterns
on four vertices.
  One of the few graph classes characterized by a pattern on four vertices is
the class of intersection graphs of rectangles that are said to be grounded on
a line. This class appears naturally in the study of intersection graphs, and
similar grounded classes have recently attracted a lot of attention.
  This paper contains three parts. First, we make a survey of grounded
intersection graph classes, summarizing all the known inclusions between these
various classes. Second, we show that the correspondence between a pattern on
four vertices and grounded rectangle graphs is not an isolated phenomenon. We
establish several other pattern characterizations for geometric classes, and
show that the hierarchy of grounded intersection graph classes is tightly
interleaved with the classes defined patterns on four vertices. We claim that
forbidden patterns are a useful tool to classify grounded intersection graphs.
Finally, we give an overview of the complexity of the recognition of classes
defined by forbidden patterns on four vertices and list several interesting
open problems.
