  Grid intersection graphs are the intersection graphs of vertical and
horizontal segments in the plane. When the bottom and respectively left
endpoints of the vertical and horizontals segments belong to a line with
negative slope, the graph is called a Stick graph. Very few results exist on
Stick graphs: only small classes of Stick graphs have been identified;
recognizing Stick graphs is an open problem; and even building examples of
graphs that are not Stick graphs is quite tricky.
  In this paper, we first prove that the complements of circle graphs and of
circular arc graphs are Stick graphs. Then, we propose two certificates
allowing to decide that a graph is not a Stick graph, and use them to build new
examples of non-Stick graphs. It turns out that these examples of non-Stick
graphs, as well as all those from literature, have long holes. We thus also
investigate the place of chordal grid intersection graphs in the hierarchy of
classes built around Stick graphs.
