The excluded minor structure theorem with planarly embedded wall

TL;DR

This paper proves that large walls in nearly embedded graphs contain large planarly embedded subwalls, which is crucial for strengthening the excluded minor theorem in graph theory.

Contribution

It establishes that large walls in nearly embedded graphs always contain large planarly embedded subwalls, advancing the understanding of graph minor structures.

Findings

Large walls contain planarly embedded subwalls in nearly embedded graphs.

Supports a stronger version of the Robertson and Seymour's excluded minor theorem.

Provides foundational details for graph minor theory applications.

Abstract

A graph is nearly embedded in a surface if it consists of graph $G_0$ that is embedded in the surface, together with a bounded number of vortices having no large transactions. It is shown that every large wall (or grid minor) in a nearly embedded graph, many rows of which intersect the embedded subgraph $G_0$ of the near-embedding, contains a large subwall that is planarly embedded within $G_0$. This result provides some hidden details needed for a strong version of the Robertson and Seymour's excluded minor theorem as presented in [K. Kawarabayashi, B. Mohar, Some recent progress and applications in graph minor theory, Graphs Combin. 23 (2007) 1-46].