Separating layered treewidth and row treewidth

TL;DR

This paper demonstrates that layered treewidth and row treewidth are not bounded functions of each other, providing explicit examples of graphs with minimal layered treewidth but arbitrarily large row treewidth.

Contribution

It proves that row treewidth cannot be bounded by any function of layered treewidth, answering an open question in graph theory.

Findings

Existence of graphs with layered treewidth 1 and arbitrarily large row treewidth

Similar results for layered pathwidth and row pathwidth

Layered treewidth and row treewidth are not mutually bounded

Abstract

Layered treewidth and row treewidth are recently introduced graph parameters that have been key ingredients in the solution of several well-known open problems. It follows from the definitions that the layered treewidth of a graph is at most its row treewidth plus 1. Moreover, a minor-closed class has bounded layered treewidth if and only if it has bounded row treewidth. However, it has been open whether row treewidth is bounded by a function of layered treewidth. This paper answers this question in the negative. In particular, for every integer $k$ we describe a graph with layered treewidth 1 and row treewidth $k$. We also prove an analogous result for layered pathwidth and row pathwidth.