The Effect of Planarization on Width

TL;DR

This paper investigates how planarization affects various graph width parameters, revealing that some parameters can grow significantly while others remain bounded after planarization.

Contribution

It demonstrates that planarization can cause unbounded growth in certain width parameters, but not in others, providing a nuanced understanding of graph width behavior.

Findings

Treewidth, pathwidth, branchwidth, clique-width, and tree-depth can increase to linear size after planarization.

Bandwidth, cutwidth, and carving width remain bounded in planarizations of bounded graphs.

Bounded degree graphs retain bounded width parameters after planarization.

Abstract

We study the effects of planarization (the construction of a planar diagram $D$ from a non-planar graph $G$ by replacing each crossing by a new vertex) on graph width parameters. We show that for treewidth, pathwidth, branchwidth, clique-width, and tree-depth there exists a family of $n$-vertex graphs with bounded parameter value, all of whose planarizations have parameter value $\Omega(n)$. However, for bandwidth, cutwidth, and carving width, every graph with bounded parameter value has a planarization of linear size whose parameter value remains bounded. The same is true for the treewidth, pathwidth, and branchwidth of graphs of bounded degree.