Blow-up structure of graphs excluding a tree or an apex-tree as a minor
Quentin Claus, Gwena\"el Joret, Cl\'ement Rambaud

TL;DR
This paper establishes new structural theorems for graphs excluding certain trees or apex-trees as minors, showing they can be embedded into products of graphs with bounded pathwidth or treewidth, refining previous bounds.
Contribution
It provides improved bounds on the structure of minor-excluding graphs, specifically relating to their containment in graph products with bounded parameters, advancing prior results.
Findings
Graphs excluding a tree as a minor are contained in a product of a graph with bounded pathwidth and a complete graph.
Graphs excluding an apex-tree as a minor are contained in a product of a graph with bounded treewidth and a large complete graph.
Bounds on the parameters are tight up to constant factors, improving previous exponential bounds.
Abstract
We prove blow-up structure theorems for graphs excluding a tree or an apex-tree as a minor. First, we show that for every -vertex tree with and radius , and every graph excluding as a minor, there exists a graph with pathwidth at most such that is contained in as a subgraph. This improves on a recent theorem of Dujmovi\'c, Hickingbotham, Joret, Micek, Morin, and Wood (2024), who proved the same result but with a larger bound on the order of the complete graph in the product. Second, we show that for every -vertex tree with , radius and maximum degree , and every graph excluding the apex-tree as a minor, where is the tree obtained by adding a universal vertex to , there exists a graph with treewidth at most such that is contained in . The…
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Taxonomy
TopicsAdvanced Graph Theory Research · Interconnection Networks and Systems · Limits and Structures in Graph Theory
