The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. In this paper we study the thinness and its variations of graph products. We show that the thinness behaves "well" in general for products, in the sense that for most of the graph products defined in the literature, the thinness of the product of two graphs is bounded by a function (typically product or sum) of their thinness, or of the thinness of one of them and the size of the other. We also show for some cases the non-existence of such a function.

中文翻译：

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产品图的细化

图的薄度是一个宽度参数，它概括了区间图的一些特性，这些特性正是薄度图。给定图的合适表示，可以在多项式时间内解决具有有界薄度的图的许多 NP 完全问题。在本文中，我们研究了图形产品的薄度及其变化。我们证明了产品的薄度通常表现得“很好”，因为对于文献中定义的大多数图产品，两个图的产品的薄度受它们的函数（通常是乘积或总和）的限制薄，或其中一个的薄和另一个的大小。我们还展示了在某些情况下不存在这样的函数。