Linear Layouts of Bipartite Planar Graphs

TL;DR

This paper studies linear layouts of bipartite planar graphs, providing improved upper bounds on the number of queues needed and establishing new lower bounds, thus advancing understanding of their structural properties.

Contribution

It offers the first non-trivial upper bound of 28 queues for bipartite planar graphs and proves that fewer than three queues are insufficient, addressing open questions in the field.

Findings

Improved upper bound of 28 queues for bipartite planar graphs

Two queues or one queue with one stack are not enough for bipartite planar graphs

Constructed 5-queue layouts for 2-degenerate quadrangulations

Abstract

A linear layout of a graph $ G $ consists of a linear order $\prec$ of the vertices and a partition of the edges. A part is called a queue (stack) if no two edges nest (cross), that is, two edges $ (v,w) $ and $ (x,y) $ with $ v \prec x \prec y \prec w $ ($ v \prec x \prec w \prec y $) may not be in the same queue (stack). The best known lower and upper bounds for the number of queues needed for planar graphs are 4 [Alam et al., Algorithmica 2020] and 42 [Bekos et al., Algorithmica 2022], respectively. While queue layouts of special classes of planar graphs have received increased attention following the breakthrough result of [Dujmovi\'c et al., J. ACM 2020], the meaningful class of bipartite planar graphs has remained elusive so far, explicitly asked for by Bekos et al. In this paper we investigate bipartite planar graphs and give an improved upper bound of 28 by refining existing techniques. In contrast, we show that two queues or one queue together with one stack do not suffice; the latter answers an open question by Pupyrev [GD 2018]. We further investigate subclasses of bipartite planar graphs and give improved upper bounds; in particular we construct 5-queue layouts for 2-degenerate quadrangulations.