Threshold tolerance graphs and their complement graphs, known as co-TT graphs, were introduced by Monma, Reed, and Trotter[24]. Building on this, Hell et al.[19] introduced the concept of negative interval. Then they proceeded to define signedinterval digraphs/ bigraphs, demonstrating their equivalence to several seemingly distinct classes of digraphs/ bigraphs. They also showed that co-TT graphs are equivalent to symmetric signed-interval digraphs, where some vertices of the digraphs have loops and others do not. We have showed that this actually solve the representation characterization problem of co-TT graphs posed by Monma, Reed and Trotter [24]. In this paper, we characterize signed-interval bigraphs and signed-interval graphs in terms of their biadjacency matrices and adjacency matrices, respectively. Moreover we emphasize on the geometric representation of signed-interval graphs, i.e. co-TT graphs. Finally, by utilizing the geometric representation of signed-interval graphs, we resolve the open problem of characterizing co-TT graphs in terms of minimal forbidden induced subgraphs, a problem initially posed by Monma, Reed, and Trotter in the same paper.

翻译：阈值容忍图及其补图，即共-TT图，由Monma、Reed和Trotter[24]引入。在此基础上，Hell等人[19]引入了负区间的概念。随后他们定义了有符号区间有向图/二分图，并证明了其与若干看似不同的有向图/二分图类等价。他们还证明了共-TT图等价于对称有符号区间有向图，其中部分顶点带自环而其余顶点则无。我们指出这实际上解决了Monma、Reed和Trotter[24]提出的共-TT图表征刻画问题。本文分别通过双邻接矩阵和邻接矩阵刻画了有符号区间二分图与有符号区间图。此外，我们重点探讨了有符号区间图（即共-TT图）的几何表示。最后，利用有符号区间图的几何表示，我们解决了Monma、Reed和Trotter在同一篇论文中提出的关于通过极小禁止诱导子图刻画共-TT图的公开问题。