  A graph $G=(V,E)$ is a geometric intersection graph if every node $v \in V$
is identified with a geometric object of some particular type, and two nodes
are adjacent if the corresponding objects intersect. Geometric intersection
graph classes have been studied from both the theoretical and practical point
of view. On the one hand, many hard problems can be efficiently solved or
approximated when the input graph is restricted to a geometric intersection
class of graphs. On the other hand, these graphs appear naturally in many
applications such as sensor networks, scheduling problems, and others.
Recently, in the context of distributed certification and distributed
interactive proofs, the recognition of graph classes has started to be
intensively studied. Different results related to the recognition of trees,
bipartite graphs, bounded diameter graphs, triangle-free graphs, planar graphs,
bounded genus graphs, $H$-minor free graphs, etc., have been obtained.
  The goal of the present work is to design efficient distributed protocols for
the recognition of relevant geometric intersection graph classes, namely
permutation graphs, trapezoid graphs, circle graphs, and polygon-circle graphs.
More precisely, for the two first classes, we give proof labeling schemes
recognizing them with logarithmic-sized certificates. For the other two
classes, we give three-round distributed interactive protocols that use
messages and certificates of size $\mathcal{O}(\log n)$. Finally, we provide
logarithmic lower-bounds on the size of the certificates on the proof labeling
schemes for the recognition of any of the aforementioned geometric intersection
graph classes.
