  The thinness of a graph is a width parameter that generalizes some properties
of interval graphs, which are exactly the graphs of thinness one. Graphs with
thinness at most two include, for example, bipartite convex graphs. Many
NP-complete problems can be solved in polynomial time for graphs with bounded
thinness, given a suitable representation of the graph. Proper thinness is
defined analogously, generalizing proper interval graphs, and a larger family
of NP-complete problems are known to be polynomially solvable for graphs with
bounded proper thinness. It is known that the thinness of a graph is at most
its pathwidth plus one. In this work, we prove that the proper thinness of a
graph is at most its bandwidth, for graphs with at least one edge. It is also
known that boxicity is a lower bound for the thinness. The main results of this
work are characterizations of 2-thin and 2-proper thin graphs as intersection
graphs of rectangles in the plane with sides parallel to the Cartesian axes and
other specific conditions. We also bound the bend number of graphs with low
thinness as vertex intersection graphs of paths on a grid ($B_k$-VPG graphs are
the graphs that have a representation in which each path has at most $k$
bends). We show that 2-thin graphs are a subclass of $B_1$-VPG graphs and,
moreover, of monotone L-graphs, and that 3-thin graphs are a subclass of
$B_3$-VPG graphs. We also show that $B_0$-VPG graphs may have arbitrarily large
thinness, and that not every 4-thin graph is a VPG graph. Finally, we
characterize 2-thin graphs by a set of forbidden patterns for a vertex order.
