On the queue-number of graphs with bounded tree-width

TL;DR

This paper proves that graphs with bounded tree-width have queue-number at most exponential in the tree-width, improving previous bounds, and constructs specific graphs demonstrating lower bounds for queue-number.

Contribution

It establishes new upper bounds on the queue-number of graphs with bounded tree-width and provides constructions showing these bounds are tight for certain cases.

Findings

Graphs with tree-width at most k have queue-number at most 2^k - 1.

These graphs have track-number at most 2^{O(k^2)}.

Constructed k-trees with queue-number at least k+1.

Abstract

A queue layout of a graph consists of a linear order on the vertices and an assignment of the edges to queues, such that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called its queue-number. We show that for each $k\geq1$, graphs with tree-width at most $k$ have queue-number at most $2^k-1$. This improves upon double exponential upper bounds due to Dujmovi\'c et al. and Giacomo et al. As a consequence we obtain that these graphs have track-number at most $2^{O(k^2)}$. We complement these results by a construction of $k$-trees that have queue-number at least $k+1$. Already in the case $k=2$ this is an improvement to existing results and solves a problem of Rengarajan and Veni Madhavan, namely, that the maximal queue-number of $2$-trees is equal to $3$.