  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.
