Two Results on Layered Pathwidth and Linear Layouts

TL;DR

This paper establishes new bounds connecting layered pathwidth with stack number and track number, advancing understanding of graph layout parameters and solving an open problem in the field.

Contribution

It proves that stack number is at most four times layered pathwidth and that graphs with track number at most three have layered pathwidth at most four, linking these parameters.

Findings

Stack number ≤ 4 × layered pathwidth

Graphs with track number ≤ 3 have layered pathwidth ≤ 4

Solves an open problem on layered pathwidth bounds

Abstract

Layered pathwidth is a new graph parameter studied by Bannister et al (2015). In this paper we present two new results relating layered pathwidth to two types of linear layouts. Our first result shows that, for any graph $G$, the stack number of $G$ is at most four times the layered pathwidth of $G$. Our second result shows that any graph $G$ with track number at most three has layered pathwidth at most four. The first result complements a result of Dujmovi\'c and Frati (2018) relating layered treewidth and stack number. The second result solves an open problem posed by Bannister et al (2015).