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.

翻译：图形的薄度是一个宽度参数, 它是一个宽度参数, 泛化一些间距图的属性, 它们是精确的薄度图。 薄度图在多数两种情况下包含, 例如, 薄度图的图形。 许多NP- 完整的问题可以在多角时间解决, 其细度是适当的图形。 适当的薄度被类似地定义, 概括适当的间距图, 以及更大范围的NP- 完整的问题, 已知对于有适当薄度图的图形来说, 是多元的。 已知, 薄度图的薄度是薄度图 。 薄度图的薄度图最多是路径 双边 。 纯度图中, 纯度图中, 纯度图中, 纯度图中, 纯度图中, 值 的值是每平面图中, 美元 。