Obstructions to Erdős-Pósa Dualities for Minors

TL;DR

This work develops a comprehensive theory of Erdős-Pósa dualities for graph minors across all proper minor-closed target classes ${\cal H}$. It introduces a finite obstruction framework ${\frak C}_{\cal H}$, proving that such counterexamples are countable and describable via wall-like parametric graphs called walloids, which admit half-integral barriers for ${\cal H}$-barriers and large modulators. The authors formulate a local structure theorem that reduces global structure to a meadow/garden/orchard progression, enabling constructive proofs of half-integral Erdős-Pósa properties for minors and a grid-like theory for ${\cal H}$-treewidth, with explicit algorithmic consequences including FPT procedures for half-integral packing and decision problems about EP-pairs. They extend the framework to topological minors, discuss limitations, and propose potential improvements to gap bounds contingent on advances in the Graph Minors Structure Theorem (GMST). Collectively, the results yield a constructive, algorithmically tractable dichotomy for Erdős-Pósa dualities in the minor order, plus a robust toolkit of walloid-based universal obstructions that unify and extend classical grid and minor theories.

Abstract

Let ${\cal G}$ and ${\cal H}$ be minor-closed graph classes. The pair $({\cal H},{\cal G})$ is an Erdős-Pósa 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 Erdős and Pósa 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 Erdős-Pósa property for minors and yields a constructive proof of Thomas' conjecture on the half-integral Erdős-Pósa 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 $2^{2^{k^{{\cal O}_h(1)}}}\cdot |G|^4\log |G|.$

Figures (31)

• Figure 1: Two wall-like graphs in $\mathfrak{F}^{1/2}_{\mathcal{H}}$ when $\mathcal{H}$ is the class of $K_{3,3}$-minor free graphs.
• Figure 2: A strict bramble of subgraphs, all obstructing membership to $\mathcal{H},$ in a member $\mathscr{W}_k$ of $\mathfrak{W}_{\mathcal{H}}.$ The blue strip at the bottom represents the large wall that constitutes the basis of $\mathscr{W}_k.$ At the top of this wall, we attach gadgets with different properties. These gadgets are, from left to right, two "handle segments" and two "crosscap segments" which model a surface. What follows are several "flower segments". These segments model parts of an obstruction $Z\in\mathsf{obs}(\mathcal{H})$ which attach to the rest of the graph through few (that is less than three) vertices. Each "flower segment" hosts $k$ disjoint copies of the same part. Finally, on the far right, there are additional "flower segments". These flowers still model parts of $Z,$ but these parts are now allowed to attach to the rest of the graph via boundaries of size larger than three (although we will show that the boundaries are still of bounded size). The gray area serves two purposes. First, it illustrates one possible way $Z$ might fit into $\mathscr{W}_k$ as a minor. Second, it indicates how to draw several copies of $Z$ in a way that no vertex of $\mathscr{W}_k$ is contained in more than two copies and that any two copies intersect. Such a family of copies of $Z,$ embedded as minors into $\mathscr{W}_k$ forms the strict bramble. See \ref{['fig_half_integral_packing']} and \ref{['lemma_half_integral_Brambles']} for a more detailed description.
• Figure 3: The iceberg provides a visual metaphor for the landscape on Erdős-Pósa dualities for (topological) minor-closed graph classes.
• Figure 4: From left to right: an elementary $3$-wall segment, an elementary $3$-handle segment, an elementary $3$-crosscap segment, and an elementary $(3,3,K_{3,3}^-,\langle y_1,y_2\rangle)$-flower segment.
• Figure 5: Left: A graph $M=Z$ and a subgraph partition $\mathcal{P}$ of $M$ containing two graphs $H_{1}$ and $H_{2}.$$H_{1}$ is drawn in the green disk while $H_{2}$ is drawn in the violet disk. The hypergraph $\mathcal{K}_{\mathcal{P}}$ contains the vertices $a,b,c,$ and $d$ and two hyper-edges $e_{1}=e_{2}=\{a,b,c,d\}.$ The figure shows an embedding of $\mathcal{K}_{\mathcal{P}}$ in the sphere. The choices for $Z,$$M,$$\mathcal{P},$ and $\eta,$ generate the embedding pair $\mathbf{p}=(\Sigma,\mathbf{B})$ for the class $\mathsf{excl}(Z)$ where $\mathbf{B}=\{\langle H_{1},\langle d,b,a,c\rangle\rangle,\langle H_{2},\langle c,a,b,d\rangle\rangle\}.$ Right: The guest walloid $\mathscr{W}^{\mathbf{p}}_3.$

Theorems & Definitions (105)

• Theorem 1.1: Main result
• Theorem 1.2: Local structure theorem (simplified)
• Theorem 1.3
• Theorem 1.4: General grid theorem
• Corollary 1
• Theorem 1.5
• Theorem 1.6
• Proposition 1
• Lemma 1
• proof