Treewidth of grid subsets

TL;DR

This paper investigates the treewidth of certain grid-like graphs and demonstrates that any vertex set separating the left and right sides of the cube induces a subgraph with high treewidth, implying limitations on partitioning such graphs into low-treewidth parts.

Contribution

The paper introduces a lower bound on the treewidth of separator subgraphs in grid graphs with diagonals and extends this to show the impossibility of partitioning the entire graph into low-treewidth components.

Findings

Any separator set in Q_n induces a subgraph with treewidth at least n/√18 - 1.

Q_n cannot be partitioned into two parts each with bounded treewidth.

The results generalize to broader classes of grid graphs with diagonals.

Abstract

Let Q_n be the graph of n times n times n cube with all non-decreasing diagonals (including the facial ones) in its constituent unit cubes. Suppose that a subset S of V(Q_n) separates the left side of the cube from the right side. We show that S induces a subgraph of tree-width at least n/sqrt{18}-1. We use a generalization of this claim to prove that the vertex set of Q_n cannot be partitioned to two parts, each of them inducing a subgraph of bounded tree-width.