A characterization of uniquely representable two-directional orthogonal ray graphs

TL;DR

This paper characterizes when two-directional orthogonal ray graphs have unique representations, linking their structure to the absence of certain subgraphs called buried subgraphs, thus extending concepts from interval graph theory.

Contribution

It introduces the concept of buried subgraphs for two-directional orthogonal ray graphs and proves their absence characterizes unique representability, a novel extension of interval graph theory.

Findings

Unique representability is characterized by the absence of buried subgraphs.

Buried subgraphs are necessary and sufficient for non-uniqueness.

The work connects orthogonal ray graphs to well-studied graph classes.

Abstract

In this paper, we provide a characterization of uniquely representable two-directional orthogonal ray graphs, which are defined as the intersection graphs of rightward and downward rays. The collection of these rays is called a representation of the graph. Two-directional orthogonal ray graphs are equivalent to several well-studied classes of graphs, including complements of circular-arc graphs with clique cover number two. Normalized representations of two-directional orthogonal ray graphs, where the positions of certain rays are determined by neighborhood containment relations, can be obtained from the normalized representations of circular-arc graphs. However, the normalized representations are not necessarily unique, even when considering only the relative positions of the rays. Recent studies indicate that two-directional orthogonal ray graphs share similar characterizations to interval graphs. Hanlon (1982) and Fishburn (1985) characterized uniquely representable interval graphs by introducing the notion of a buried subgraph. Following their characterization, we define buried subgraphs of two-directional orthogonal ray graphs and prove that their absence is a necessary and sufficient condition for a graph to be uniquely representable.