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arXiv · 1602.08669

Interval k-Graphs and Orders

David E. Brown·Breeann M. Flesch·Larry J. Langley
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Abstract

An interval $k$-graph is the intersection graph of a family $\mathcal{I}$ of intervals of the real line partitioned into at most $k$ classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we discuss the interval $k$-graphs that are the incomparability graphs of orders; i.e., cocomparability interval $k$-graphs or interval $k$-orders. Interval $2$-orders have been characterized in many ways, but we show that analogous characterizations do not carry over to interval $k$-orders, for $k > 2$. We describe the structure of interval $k$-orders, for any $k$, characterize the interval $3$-orders (cocomparability interval $3$-graphs) via one forbidden suborder (subgraph), and state a conjecture for interval $k$-orders (any $k$) that would characterize them via two forbidden suborders.

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David E. Brown, Breeann M. Flesch, Larry J. Langley. 2016-02-28. Interval k-Graphs and Orders. https://arxiv.org/abs/1602.08669

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