Rooted grid minors

TL;DR

This paper proves that in highly connected graphs with a large tangle and a vertex set Z that cannot be separated from the tangle by small cuts, there exists a large grid minor containing Z on its outside, aiding future research.

Contribution

It establishes a new lemma linking large tangles, vertex sets, and grid minors, enhancing understanding of graph structure and connectivity.

Findings

Large grid minors exist under specified tangle and vertex set conditions.

Vertices in Z are positioned on the outside of the grid minor.

The result facilitates further structural graph theory research.

Abstract

Intuitively, a tangle of large order in a graph is a highly-connected part of the graph, and it is known that if a graph has a tangle of large order then it has a large grid minor. Here we show that for any k, if G has a tangle of large order and Z is a set of vertices of cardinality k that cannot be separated from the tangle by any separation of order less than k, then G has a large grid minor containing Z, in which the members of Z all belong to the outside of the grid. This is a lemma for use in a later paper.