Excluding disjoint Kuratowski graphs
Neil Robertson, Paul Seymour

TL;DR
This paper investigates graphs that exclude certain disjoint unions of Kuratowski graphs as minors, showing that such graphs can be simplified by removing a bounded number of vertices to become surface-embeddable without these minors.
Contribution
It establishes a structural characterization of graphs excluding disjoint Kuratowski minors, extending the understanding of graph minors and surface embeddings.
Findings
Graphs excluding disjoint Kuratowski minors can be made surface-embeddable after removing a bounded set of vertices.
The result generalizes previous minor exclusion theorems to disjoint unions of Kuratowski graphs.
Provides a method to identify a small vertex set to simplify complex graphs for topological analysis.
Abstract
A graph is a ``-Kuratowski graph'' if it has exactly components, each isomorphic to or to . We prove that if a graph contains no -Kuratowski graph as a minor,then there is a set of boundedly many vertices such that can be drawn in a (possibly disconnected) surface in which no -Kuratowski graph can be drawn.
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Advanced Graph Theory Research · graph theory and CDMA systems
