The Queue-Number of Posets of Bounded Width or Height

Abstract

Heath and Pemmaraju [9] conjectured that the queuenumber of a poset is bounded by its width and if the poset is planar then also by its height. We show that there are planar posets whose queue-number is larger than their height, refuting the second conjecture. On the other hand, we show that any poset of width 2 has queue-number at most 2, thus confirming the first conjecture in the first non-trivial case. Moreover, we improve the previously best known bounds and show that planar posets of width w have queue-number at most 3w − 2 while any planar poset with 0 and 1 has queue-number at most its width.

Planar Graphs have Bounded Queue-Number

“…Heath and Pemmaraju [64] conjectured that the queue-number of a planar poset is at most its height (the maximum number of pairwise comparable elements). This was disproved by Knauer, Micek, and Ueckerdt [71] who presented a poset of height 2 and queue-number 4. Theorem 1 and results of Knauer, Micek and Ueckerdt imply that planar posets of height h have queue-number O(h); see Theorem 6 in [71].…”

“…Heath and Pemmaraju [64] conjectured that the queue-number of a planar poset is at most its height (the maximum number of pairwise comparable elements). This was disproved by Knauer, Micek, and Ueckerdt [71] who presented a poset of height 2 and queue-number 4.…”

“…Theorem 1 and results of Knauer, Micek and Ueckerdt imply that planar posets of height h have queue-number O(h); see Theorem 6 in [71]. Heath and Pemmaraju [64] also conjecture that every poset of width w (the maximum number of pairwise incomparable elements) has queue-number at most w. The best known upper bounds are O(w 2 ) for general posets and 3w − 2 for planar posets [71].…”

Lazy Queue Layouts of Posets

“…Heath and Pemmaraju [11] made a step towards settling the conjecture by providing a linear upper bound of 4w − 1 on the queue number of planar posets of width w. This bound was recently improved to 3w − 2 by Knauer, Micek, and Ueckerdt [15], who also gave a planar poset whose queue number is exactly w, thus establishing a lower bound. Furthermore, they investigated (non-planar) posets of width 2, and proved that their queue number is at most 2.…”

“…In Theorem 5 in Appendix B.1, we present a poset and a linear extension of it which yields a rainbow of size w 2 . Knauer et al [15] studied a special type of linear extensions, called lazy, for posets of width-2 to show that their queue number is at most 2. Thus, it is tempting to generalize and analyze lazy linear extensions for posets of width w > 2.…”