The Local Queue Number of Graphs with Bounded Treewidth

TL;DR

This paper introduces the concept of local queue number for graphs, explores its relationship with graph density and treewidth, and establishes bounds for graphs of bounded treewidth, with implications for planar graphs.

Contribution

It defines the local queue number, provides bounds for graphs with bounded treewidth, and shows the tightness of these bounds for certain cases.

Findings

Graphs of treewidth k have local queue number at most k+1.

The bound is tight for k=2.

Maximum local queue number among planar graphs is 3 or 4.

Abstract

A queue layout of a graph $G$ consists of a vertex ordering of $G$ and a partition of the edges into so-called queues such that no two edges in the same queue nest, i.e., have their endpoints ordered in an ABBA-pattern. Continuing the research on local ordered covering numbers, we introduce the local queue number of a graph $G$ as the minimum $\ell$ such that $G$ admits a queue layout with each vertex having incident edges in no more than $\ell$ queues. Similarly to the local page number [Merker, Ueckerdt, GD'19], the local queue number is closely related to the graph's density and can be arbitrarily far from the classical queue number. We present tools to bound the local queue number of graphs from above and below, focusing on graphs of treewidth $k$. Using these, we show that every graph of treewidth $k$ has local queue number at most $k+1$ and that this bound is tight for $k=2$, while a general lower bound is $\lceil k/2\rceil+1$. Our results imply, inter alia, that the maximum local queue number among planar graphs is either 3 or 4.