Queue Layouts of Two-Dimensional Posets

TL;DR

This paper investigates queue layouts of two-dimensional posets, disproves a conjecture for these posets by providing a counterexample, and establishes improved upper bounds on their queue numbers.

Contribution

It constructs a two-dimensional poset with width w > 2 having queue number 2(w - 1), disproving the conjecture, and improves the upper bound to w(w+1)/2.

Findings

Counterexample with queue number 2(w - 1) for width w > 2

Disproof of the conjecture for two-dimensional posets

Improved upper bound of w(w+1)/2 on queue number

Abstract

The queue number of a poset is the queue number of its cover graph when the vertex order is a linear extension of the poset. Heath and Pemmaraju conjectured that every poset of width $w$ has queue number at most $w$. The conjecture has been confirmed for posets of width $w=2$ and for planar posets with $0$ and $1$. In contrast, the conjecture has been refused by a family of general (non-planar) posets of width $w>2$. In this paper, we study queue layouts of two-dimensional posets. First, we construct a two-dimensional poset of width $w > 2$ with queue number $2(w - 1)$, thereby disproving the conjecture for two-dimensional posets. Second, we show an upper bound of $w(w+1)/2$ on the queue number of such posets, thus improving the previously best-known bound of $(w-1)^2+1$ for every $w > 3$.