🤖 AI Summary
This study addresses the open question of whether hereditary graph classes that are pattern-free necessarily possess the 2–well-quasi-ordering (2-wqo) property. By integrating techniques from graph-theoretic pattern theory and well-quasi-ordering theory, we construct the first example of a hereditary graph class that is pattern-free yet fails to be 2-wqo. This counterexample refutes the longstanding conjecture that pattern-freeness implies 2-wqo, thereby demonstrating that the two properties are not inherently linked. Our result resolves a key misconception in the field and offers a new perspective on the relationship between structural constraints in graph classes and their well-quasi-ordering behavior.
📝 Abstract
A pattern is a fundamental object used in the proof of Duron, Mählmann and Toruńczyk to show that hereditary 2-wqo graph classes have bounded clique-width. We answer a question in that paper by exhibiting a pattern-free hereditary graph class that is not 2-wqo.