Treewidth Bounds for Planar Graphs Using Three-Sided Brambles

TL;DR

This paper introduces a new method using three-sided brambles called nets to efficiently estimate the treewidth of planar graphs, improving existing bounds and providing a polynomial-time algorithm for bounds computation.

Contribution

It defines nets as a generalization of brambles, characterizes minimal covers, and develops an $O(n^3)$ algorithm to approximate the bramble number and treewidth of planar graphs.

Findings

Provides bounds relating net order to bramble number and treewidth.

Improves previous bounds on grid minors and treewidth estimates.

Offers an efficient polynomial-time algorithm for planar graph analysis.

Abstract

Square grids play a pivotal role in Robertson and Seymour's work on graph minors as planar obstructions to small treewidth. We introduce a three-sided bramble in a plane graph called a net, which generalizes the standard bramble of crosses in a square grid. We then characterize any minimal cover of a net as a tree drawn in the plane. We use nets in an $O(n^3)$ time algorithm that computes both upper and lower bounds on the bramble number (hence treewidth) of any planar graph. Let $G$ be a planar graph, $BN(G)$ be its bramble number and $\lambda(G)$ be the largest order of any net in a subgraph of $G$. Our algorithm outputs a constant, $KB$, so that $\lambda(G)/4 \leq KB \leq BN(G)\leq 4KB \leq 4\lambda(G)$. Let $s(G)$ be the size of a side of the largest square grid minor of $G$. Smith (2015) has shown that $\lambda(G) \geq s(G)$. Our upper bound improves that of Grigoriev (2011) when $\lambda(G)\leq (5/4)s(G)$. We correct a lower bound of Bodlaender, Grigoriev and Koster (2008) to $s(G)/5$ (instead of $s(G)/4$) and thus the lower bound of $\lambda(G)/4$ on our approximation is an improvement.