Double-threshold graphs are defined in terms of two real thresholds that break the real line into three regions, alternating as NO-YES-NO. If real ranks can be assigned to the vertices of a graph in such a way that two vertices are adjacent iff the sum of their ranks lies in the YES region, then that graph is a double-threshold graph with respect to the given set of thresholds. We demonstrate that all double-threshold graphs are permutation graphs. As a partial converse, we show that every bipartite permutation graph has a balanced double-threshold representation. That is, the vertices with negative rank form one part of the bipartition, those with positive rank the other part.

双阈值图是根据两个实际阈值定义的，这些阈值将实际线分成三个区域，交替为 NO-YES-NO。如果可以将实际秩分配给图的顶点，使得两个顶点相邻且仅当它们的秩和位于 YES 区域时，则该图是关于给定阈值集的双阈值图. 我们证明所有双阈值图都是置换图。作为部分相反，我们表明每个二分置换图都有一个平衡的双阈值表示。也就是说，负秩的顶点构成二分的一部分，正秩的顶点构成另一部分。