Interval-Like Graphs and Digraphs

TL;DR

This paper unifies various graph and digraph classes related to interval graphs through a common framework involving loops, min orderings, and signed-interval digraphs, providing new geometric and ordering characterizations.

Contribution

It introduces signed-interval digraphs as a unifying class and characterizes multiple graph classes via min orderings and geometric properties.

Findings

Unified several graph classes under signed-interval digraphs

Characterized classes using min orderings and geometric methods

Connected classes like co-TT graphs to signed-interval digraphs

Abstract

We unify several seemingly different graph and digraph classes under one umbrella. These classes are all broadly speaking different generalizations of interval graphs, and include, in addition to interval graphs, also adjusted interval digraphs, threshold graphs, complements of threshold tolerance graphs (known as `co-TT' graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray graphs. (The last three classes coincide, but have been investigated in different contexts.) This common view is made possible by introducing loops. We also show that all the above classes are united by a common ordering characterization, the existence of a min ordering. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, and show that they are precisely the digraphs that are characterized by the existence of a min ordering. We also offer an alternative geometric characterization of these digraphs. For most of the above example graph and digraph classes, we show that they are exactly those signed-interval digraphs that satisfy a suitable natural restriction on the digraph, like having all loops, or having a symmetric edge-set, or being bipartite. (For instance co-TT graphs are precisely those signed-interval digraphs that have each edge symmetric.) We also offer some discussion of recognition algorithms and characterizations, saving the details for future papers.