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.

翻译：图 $ G $ 的线性布局由顶点的一个线性序 $\prec$ 和边的一个划分组成。若两条边 $ (v,w) $ 与 $ (x,y) $ 满足 $ v \prec x \prec y \prec w $（$ v \prec x \prec w \prec y $），则它们不能属于同一个队列（栈），否则称这两条边嵌套（交叉）。已知平面图所需队列数量的最佳下界和上界分别为 4 [Alam 等, Algorithmica 2020] 和 42 [Bekos 等, Algorithmica 2022]。尽管在 [Dujmović 等, J. ACM 2020] 的突破性结果后，特殊平面图类的队列布局研究日益受到关注，但具有重要意义的二分平面图类至今仍难以突破，Bekos 等人曾明确提出这一问题。本文研究二分平面图，通过改进现有技术将上界优化至 28。相反地，我们证明两个队列或一个队列加一个栈均不足以应对此类图；后者回答了 Pupyrev [GD 2018] 的一个开放问题。我们进一步研究二分平面图的子类并给出了改进的上界：特别地，为 2-退化四边形图构造了 5-队列布局。