Uniquely orderable interval graphs

TL;DR

This paper explores the conditions under which an interval graph corresponds to a unique interval order, extending Fishburn's finite case characterization to infinite graphs and explaining the limitations of this extension using reverse mathematics.

Contribution

It provides a new proof that the finite characterization of uniquely orderable interval graphs applies to infinite graphs and analyzes why this does not extend via compactness.

Findings

Finite and infinite connected interval graphs share the same unique orderability characterization.

The extension from finite to infinite graphs cannot be achieved through compactness, explained via reverse mathematics.

The paper offers a new proof technique for the characterization of uniquely orderable interval graphs.

Abstract

Interval graphs and interval orders are deeply linked. In fact, edges of an interval graphs represent the incomparability relation of an interval order, and in general, of different interval orders. The question about the conditions under which a given interval graph is associated to a unique interval order (up to duality) arises naturally. Fishburn provided a characterisation for uniquely orderable finite connected interval graphs. We show, by an entirely new proof, that the same characterisation holds also for infinite connected interval graphs. Using tools from reverse mathematics, we explain why the characterisation cannot be lifted from the finite to the infinite by compactness, as it often happens.