Obstructions to Erd\H{o}s-P\'osa Dualities for Minors
Christophe Paul, Evangelos Protopapas, Dimitrios M. Thilikos and, Sebastian Wiederrecht

TL;DR
This paper characterizes obstructions to Erdős-Pósa dualities for minors, proving finiteness of counterexamples, describing their structure via grid-like graphs, and providing algorithms for packing and hitting minors.
Contribution
It offers a complete characterization of minimal EP-counterexamples for minor-closed classes and constructs algorithms for packing and hitting minors with explicit bounds.
Findings
Finite set of EP-counterexamples for each ${ m H}$.
Complete structural description of counterexamples as minors of grid-like graphs.
Constructive algorithms with explicit bounds for packing and hitting minors.
Abstract
Let and be minor-closed graph classes. The pair is an Erd\H{o}s-P\'osa pair (EP-pair) if there is a function where, for every and every either has pairwise vertex-disjoint subgraphs not belonging to or there is a set where and The classic result of Erd\H{o}s and P\'osa says that if is the class of forests, then is an EP-pair for every . The class is an EP-counterexample for if is minimal with the property that is not an EP-pair. We prove that for every the set of all EP-counterexamples for is finite. In particular, we provide a complete characterization of for every and…
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Taxonomy
TopicsAdvanced Algebra and Geometry · Mathematical Analysis and Transform Methods · Holomorphic and Operator Theory
