Obstruction characterization of co-TT graphs

TL;DR

This paper characterizes co-TT graphs through their matrix representations and geometric models, ultimately solving the open problem of their minimal forbidden induced subgraphs.

Contribution

It provides a complete characterization of co-TT graphs using matrix and geometric representations, resolving a longstanding open problem.

Findings

Characterization of signed-interval bigraphs and graphs via matrices

Geometric representation of co-TT graphs

Identification of minimal forbidden induced subgraphs for co-TT graphs

Abstract

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.