Threshold tolerance graphs and their complement graphs ( known as co-TT graphs) were introduced by Monma, Reed and Trotter[24]. Introducing the concept of negative interval Hell et al.[19] defined signed-interval bigraphs/digraphs and have shown that they are equivalent to several seemingly different classes of bigraphs/digraphs. They have also shown that co-TT graphs are equivalent to symmetric signed-interval digraphs. In this paper we characterize signed-interval bigraphs and signed-interval graphs respectively in terms of their biadjacency matrices and adjacency matrices. Finally, based on the geometric representation of signed-interval graphs we have setteled the open problem of forbidden induced subgraph characterization of co-TT graphs posed by Monma, Reed and Trotter in the same paper.

翻译：阈值容差图及其补图（称为co-TT图）由Monma、Reed和Trotter[24]引入。Hell等人[19]通过引入负区间概念，定义了符号区间双图/有向图，并证明了它们与若干看似不同的双图/有向图类等价。他们还证明了co-TT图与对称符号区间有向图等价。本文分别通过双邻接矩阵和邻接矩阵刻画了符号区间双图和符号区间图。最后，基于符号区间图的几何表示，我们解决了Monma、Reed和Trotter在同一论文中提出的关于co-TT图禁止诱导子图刻画的开放问题。