Interval-Like Graphs and 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, 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 reflexive relationships (loops) into the analysis. 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 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 a loop on every vertex, 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 future work on recognition algorithms and characterizations.

Strongly chordal digraphs and $Γ$-free matrices

“…Nevertheless, it is possible to obtain useful generalizations for digraphs that are neither reflexive nor irreflexive. This is done, for example, in [14,15], where general digraphs (that have some vertices with loops and others without) avoiding * 1 1 0 are investigated and found a useful unification of interval graphs, adjusted interval digraphs, two-dimensional orthogonal ray graphs (alias interval containment digraphs), and complements of threshold tolerance graphs. Another situation where it is fruitful to admit some vertices with loops and others without loops is the sub-ject of the next section; the class of graphs investigated there unifies reflexive strongly chordal graphs and irreflexive chordal bigraphs, and introduces a whole new class of well structured graphs.…”