A more accurate view of the Flat Wall Theorem

TL;DR

This paper enhances the Flat Wall Theorem by introducing new combinatorial concepts like wall homogeneity and regularity, aiming to improve the applicability of the irrelevant vertex technique in algorithms.

Contribution

It proposes two variants of the Flat Wall Theorem and introduces new concepts to support algorithmic applications involving flat walls.

Findings

Two variants of the Flat Wall Theorem are suggested.

New concepts of wall homogeneity and regularity are introduced.

The framework facilitates the use of the irrelevant vertex technique.

Abstract

We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in flat walls. All proposed concepts and results aim at facilitating the use of the irrelevant vertex technique in future algorithmic applications.