Interval k-Graphs and Orders
David E. Brown, Breeann M. Flesch, Larry J. Langley

TL;DR
This paper explores the structure and characterization of interval k-graphs and interval k-orders, focusing on their properties as cocomparability graphs and identifying key forbidden suborders for k=3.
Contribution
It characterizes interval 3-orders using a single forbidden suborder and proposes a conjecture for characterizing interval k-orders with two forbidden suborders.
Findings
Interval 2-orders have multiple characterizations.
Characterization of interval 3-orders via one forbidden suborder.
A conjecture for characterizing interval k-orders with two forbidden suborders.
Abstract
An interval -graph is the intersection graph of a family of intervals of the real line partitioned into at most classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we discuss the interval -graphs that are the incomparability graphs of orders; i.e., cocomparability interval -graphs or interval -orders. Interval -orders have been characterized in many ways, but we show that analogous characterizations do not carry over to interval -orders, for . We describe the structure of interval -orders, for any , characterize the interval -orders (cocomparability interval -graphs) via one forbidden suborder (subgraph), and state a conjecture for interval -orders (any ) that would characterize them via two forbidden suborders.
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