Excluding a full grid minor

TL;DR

This paper characterizes graphs that do not contain a full grid minor of the form Z x Z, providing structural insights and a tree-decomposition-based description for such graphs.

Contribution

It introduces a new characterization of graphs excluding Z x Z grid minors, extending previous work on N x Z grid minors.

Findings

Characterization of graphs without Z x Z grid minors

Structural description using tree-decompositions

Extension of Halin's characterization to Z x Z grid minors

Abstract

In this paper we characterise the graphs containing a $\mathbb{Z} \times \mathbb{Z}$ grid minor in a similar way as it has been done by Halin for graphs with an $\mathbb{N} \times \mathbb{Z}$ grid minor. Using our characterisation, we describe the structure of graphs without $\mathbb{Z} \times \mathbb{Z}$ grid minors in terms of tree-decompositions.