The \emph{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. \emph{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. The complexity of recognizing 2-thin and proper 2-thin graphs is still open. In this work, we present characterizations of 2-thin and proper 2-thin graphs as intersection graphs of rectangles in the plane, as vertex intersection graphs of paths on a grid (VPG graphs), and by forbidden ordered patterns. We also prove that independent 2-thin graphs are exactly the interval bigraphs, and that proper independent 2-thin graphs are exactly the bipartite permutation graphs. Finally, we take a step towards placing the thinness and its variations in the landscape of width parameters, by upper bounding the proper thinness in terms of the bandwidth.

翻译：图的*薄度*是一种宽度参数，它推广了区间图（即薄度为1的图）的若干性质。薄度不超过2的图包括例如二分凸图。在给定图合适表示的前提下，许多NP完全问题可在多项式时间内求解薄度有界的图。*真薄度*的定义类似，它推广了真区间图，并且已知更多NP完全问题对于真薄度有界的图具有多项式可解性。2-薄图和真2-薄图的识别复杂度问题仍然开放。本文给出了2-薄图和真2-薄图的三类刻画：作为平面中矩形交图、作为网格上路径的顶点交图（VPG图）以及通过禁止有序模式。我们还证明了独立2-薄图恰为区间二分图，真独立2-薄图恰为二分置换图。最后，通过用带宽给出真薄度的上界，我们向将薄度及其变种置于宽度参数谱系中迈出了一步。