Quickly excluding a non-planar graph
Ken-ichi Kawarabayashi, Robin Thomas, Paul Wollan

TL;DR
This paper presents a new, constructive, and polynomial-time method to quickly determine if a graph is non-planar by leveraging a simplified proof of a key structural theorem in graph theory, with explicit constants.
Contribution
It provides a shorter, constructive proof of a fundamental graph minor structure theorem using only prior results and textbooks, enabling polynomial-time algorithms.
Findings
The proof is shorter and more accessible than previous ones.
It yields explicit constants for the structure theorem.
The approach allows for polynomial-time algorithms to exclude non-planar graphs.
Abstract
A cornerstone theorem in the Graph Minors series of Robertson and Seymour is the result that every graph with no minor isomorphic to a fixed graph has a certain structure. The structure can then be exploited to deduce far-reaching consequences. The exact statement requires some explanation, but roughly it says that there exist integers depending on only such that and for every grid minor of the graph has a a -near embedding in a surface that does not embed in such a way that a substantial part of is embedded in . Here a -near embedding means that after deleting at most vertices the graph can be drawn in without crossings, except for local areas of non-planarity, where crossings are permitted, but at most of these areas are attached to the rest of the graph by four or more vertices and…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Complexity and Algorithms in Graphs
