Two of the authors proved a well-known conjecture of K. Wagner, that in any infinite set of finite graphs there are two graphs so that one is a minor of the other. A key lemma was a theorem about the structure of finite graphs that have no \(K_n\) minor for a fixed integer \(n\). Here, the authors obtain an infinite analog of this lemma--a structural condition on a graph, necessary and sufficient for it not to contain a \(K_n\) minor, for any fixed infinite cardinal \(n\).

Readership

Research mathematicians in infinite graph theory.

Table of Contents

• Introduction
• Dissections
• Havens and minors
• Clustered havens of order \(\aleph _0\)
• The easy halves
• Divisions
• Long divisions
• Robust divisions
• Limited dissections
• Excluding the half-grid
• Excluding \(K_{\aleph _0}\)
• Dissections and tree-decompositions
• Topological trees
• Well-founded trees
• Well-founded tree-decompositions