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.

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2细图和适当2细图的相交模型和禁止模式表征

图的稀疏度是一个宽度参数，可以概括间隔图的某些属性，而间隔图恰好是稀疏度图。薄度至多为两个的图包括例如二分凸图。给定图的适当表示形式，对于有界稀疏图，可以在多项式时间内解决许多NP完全问题。相似地，定义适当的稀疏性，将适当的间隔图泛化，并且已知对于有界的适当稀疏性图，可以解决多项式较大的NP-完全问题。众所周知，图的稀疏度最多是其路径宽度加1。在这项工作中，我们证明了图的适当稀疏度最多是它的带宽，对于具有至少一个边的图。众所周知，方块度是薄度的下限。这项工作的主要结果是将2稀薄和2稀薄图的特征刻画为平面中矩形的交点图，边与笛卡尔轴平行，并且具有其他特定条件。我们还限制了薄度较低的图的折弯数作为网格上路径的顶点交点图（$ B_k $ -VPG图是表示每个路径最多具有$ k $折弯的图形）。我们显示2稀疏图是$ B_1 $ -VPG图的子类，而且是单调L-图的子类，而3稀疏图是$ B_3 $ -VPG图的子类。我们还显示了$ B_0 $ -VPG图可能具有任意大的稀疏度，并且并非每4个瘦图都是VPG图。最后，我们通过一组针对顶点顺序的禁止模式来表征2个稀疏图。