Obstructions to Erdős-Pósa Dualities for Minors
(2407.09671)Abstract
Let ${\cal G}$ and ${\cal H}$ be minor-closed graph classes. The pair $({\cal H},{\cal G})$ is an Erd\H{o}s-P\'osa pair (EP-pair) if there is a function $f$ where, for every $k$ and every $G\in{\cal G},$ either $G$ has $k$ pairwise vertex-disjoint subgraphs not belonging to ${\cal H},$ or there is a set $S\subseteq V(G)$ where $|S|\leq f(k)$ and $G-S\in{\cal H}.$ The classic result of Erd\H{o}s and P\'osa says that if $\mathcal{F}$ is the class of forests, then $({\cal F},{\cal G})$ is an EP-pair for every ${\cal G}$. The class ${\cal G}$ is an EP-counterexample for ${\cal H}$ if ${\cal G}$ is minimal with the property that $({\cal H},{\cal G})$ is not an EP-pair. We prove that for every ${\cal H}$ the set $\mathfrak{C}{\cal H}$ of all EP-counterexamples for ${\cal H}$ is finite. In particular, we provide a complete characterization of $\mathfrak{C}{\cal H}$ for every ${\cal H}$ and give a constructive upper bound on its size. Each class ${\cal G}\in \mathfrak{C}{\cal H}$ can be described as all minors of a sequence of grid-like graphs $\langle \mathscr{W}{k} \rangle{k\in \mathbb{N}}.$ Moreover, each $\mathscr{W}{k}$ admits a half-integral packing: $k$ copies of some $H\not\in{\cal H}$ where no vertex is used more than twice. This gives a complete delineation of the half-integrality threshold of the Erd\H{o}s-P\'osa property for minors and yields a constructive proof of Thomas' conjecture on the half-integral Erd\H{o}s-P\'osa property for minors (recently confirmed, non-constructively, by Liu). Let $h$ be the maximum size of a graph in ${\cal H}.$ For every class ${\cal H},$ we construct an algorithm that, given a graph $G$ and a $k,$ either outputs a half-integral packing of $k$ copies of some $H \not\in {\cal H}$ or outputs a set of at most ${2{k{\cal O}h(1)}}$ vertices whose deletion creates a graph in ${\cal H}$ in time $2{2{k{{\cal O}h(1)}}}\cdot |G|4\log |G|.$
Overview
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The paper expands the classic Erdős-Pósa theorem to minor-closed graph classes, providing a complete characterization and finite obstruction sets for graph classes that do not adhere to the Erdős-Pósa property.
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A novel algorithm is introduced for minor-closed graph classes that can determine either a half-integral packing of subgraphs or a bounded hitting set, running efficiently in time complexity proportional to the input size.
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The paper constructively proves and generalizes Thomas' conjecture on half-integral Erdős-Pósa properties, particularly for graph classes with planar feature obstructions, thereby creating new pathways for combinatorial optimization.
Obstructions to Erdős-Pósa Dualities for Minors
This research investigates Erdős-Pósa-type dualities for minors in graph theory, specifically focusing on the identification and characterization of minor-closed graph classes and their properties. The foundational Erdős-Pósa theorem connects hitting (vertex cover) and packing (disjoint subgraphs) problems for cycles in graphs. This paper rigorously explores such dualities beyond cycles, extending to various minor-closed classes and offering both theoretical insights and algorithmic results.
Key Contributions
1. Complete Characterization of EP-Counterexamples
The authors define a pair of graph classes ((H, G)) as an Erdős-Pósa pair (EP-pair) if there exists a function (f) such that for every (k) and graph (G \in G), the graph either contains (k) vertex-disjoint subgraphs not in (H) or has a vertex set (S) of size at most (f(k)) such that (G - S \in H). A minor-closed class (G) is an EP-counterexample for (H) if it is minimal with the property that ((H, G)) is not an EP-pair.
The paper offers a full characterization for every minor-closed graph class (H) and constructs a finite obstruction set indicating the bounds for Erdős-Pósa properties. This involves demonstrating that every class (G) can be described through the minors of a specific sequence of grid-like graphs and is capable of half-integral packings—aligning multiple copies of some (H) such that no vertex repeats more than twice.
2. Algorithmic Results
The paper introduces a novel algorithm for handling minor-closed graph classes, which, given a graph (G) and integer (k), either outputs a half-integral packing of (k) copies or a hitting set whose size is bounded by a computable function depending on (k). The algorithms run in (O(2{O(k)} \log |G|)) time and are directly applicable to several combinatorial optimization problems involving graph minors.
3. Proof and Generalization of Thomas' Conjecture
Thomas' conjecture on half-integral Erdős-Pósa properties for minors, recently confirmed by Liu, is proven constructively in this paper. Specifically, the authors show that any graph class whose obstructions have specific planar features must obey the half-integral Erdős-Pósa property—a significant extension of the conjecture with constructive proofs and polynomial bounds.
Theoretical Implications
This research enriches the theory of graph minors by providing:
- Finite descriptions of classes violating the Erdős-Pósa property.
- Minor-monotone parametrizations intricately tied to grid-like structures characterizing dualities within graph classes.
- The identification of additional graph classes named "walloids" representing generalizations of traditional grid graphs used to construct deeper and more varied obstructions.
Speculation on Future Developments
Theoretical advances from this research could inspire further exploration into:
- Stronger parametrizations for non-minor-closed properties.
- Enhanced algorithms refining bounds associated with half-integral packings.
- Applications expanding beyond graph theory to network design, computational biology, and large-scale data structures.
The intricate relationship between minor packs and hitting sets described here opens possibilities for solving complex graph embedding problems algorithmically, promising extensive methodological overflow into practical computational tasks reliant on robust combinatorial optimization.
Conclusion
"Obstructions to Erdős-Pósa Dualities for Minors" thoroughly advances our understanding of the intricate network of dualities for minor-closed graph classes. The blend of theoretical rigor, algorithmic innovation, and comprehensive characterization elucidates Erdős-Pósa properties and strengthens foundational combinatorial frameworks necessary for tackling complex structured graph challenges.
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