Chvatal and Hammer defined a graph to be a threshold graph if every vertex v has a real rank r(v) such that two vertices v and w are adjacent precisely when r(v) + r(w) ≥ 0. We extend this notion: we define a graph to be a k-threshold graph if every vertex v has a real rank r(v) and there exist k real numbers called thresholds such that two vertices v and w are adjacent precisely when r(v) + r(w) is greater than or equal to an odd number of thresholds. The 1-threshold graphs are precisely the threshold graphs of Chvatal and Hammer. The class of 2-threshold graphs is intermediate between the class of bipartite permutation graphs and the class of permutation graphs. We will report on graph classes such that few thresholds suffice, classes requiring many thresholds, and show an upper bound on the number of thresholds for all graphs on n vertices.

中文翻译：

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多阈值图

Chvatal 和 Hammer 将一个图定义为阈值图，如果每个顶点 v 都有一个实秩 r(v) 使得两个顶点 v 和 w 在 r(v) + r(w) ≥ 0 时精确相邻。 我们扩展了这个概念：如果每个顶点 v 都有一个实秩 r(v) 并且存在 k 个实数称为阈值，使得两个顶点 v 和 w 在 r(v) + r( w) 大于或等于奇数个阈值。1-threshold 图正是 Chvatal 和 Hammer 的阈值图。2-阈值图类介于二部置换图类和置换图类之间。我们将报告很少阈值就足够的图类，需要很多阈值的类，并显示 n 个顶点上所有图的阈值数量的上限。