On powers of interval graphs and their orders

TL;DR

This paper investigates the properties of powers of interval graphs, demonstrating how interval representations can be extended from one power to the next, and explores the limitations of similar extensions for trapezoid graphs.

Contribution

It extends Raychaudhuri's 1987 result by showing that interval representations of graph powers can be consistently extended, and clarifies the limitations for trapezoid graphs.

Findings

Interval representations of $G^{k-1}$ can be extended to $G^k$ with preserved endpoint orders.

The extension property holds for unit interval graphs.

The extension property does not hold for trapezoid graphs.

Abstract

It was proved by Raychaudhuri in 1987 that if a graph power $G^{k-1}$ is an interval graph, then so is the next power $G^k$. This result was extended to $m$-trapezoid graphs by Flotow in 1995. We extend the statement for interval graphs by showing that any interval representation of $G^{k-1}$ can be extended to an interval representation of $G^k$ that induces the same left endpoint and right endpoint orders. The same holds for unit interval graphs. We also show that a similar fact does not hold for trapezoid graphs.