Packing Topological Minors Half-Integrally

TL;DR

The paper addresses the half-integral Erdős–Pósa property for packing and covering $H$-topological minors in graphs, establishing that for every fixed $H$ there is a function $f$ such that either $G$ half-integrally packs $k$ ${ m R}$-compatible subdivisions of $H$ or a hitting set of size at most $f(k)$ hits all such subdivisions.The authors develop a comprehensive framework combining tangles, clique minors, nearly embedded structures, and a rich surface/vortex/pseudo-embedding apparatus to handle the topological minor setting beyond planarity, with explicit handling of root constraints via prescribed vertex subsets ${ m R}$.Key contributions include (i) a full proof of the half-integral Erdős–Pósa property for $H$-topological minors, (ii) corollaries extending to structure theorems and partition/embedding results for $H$-minor and topological-minor free graph classes, and (iii) a machinery to transfer results from $H$-subdivisions to $kH$-subdivisions in broader graph classes while preserving tight connections to $H$.The results provide a unified approach that generalizes existing minor/topological-minor theory and yields consequences such as partitioning bounds, sparse universal graphs for apex-minor-free families, and a robust link between half-integral packing and hitting-sets in a broad range of graph classes.

Abstract

The packing problem and the covering problem are two of the most general questions in graph theory. The Erdős-Pósa property characterizes the cases when the optimal solutions of these two problems are bounded by functions of each other. Robertson and Seymour proved that when packing and covering $H$-minors for any fixed graph $H$, the planarity of $H$ is equivalent to the Erdős-Pósa property. Thomas conjectured that the planarity is no longer required if the solution of the packing problem is allowed to be half-integral. In this paper, we prove that this half-integral version of Erdős-Pósa property holds for packing and covering $H$-topological minors, for any fixed graph $H$, which easily implies Thomas' conjecture. In fact, we prove an even stronger statement in which those topological minors are rooted at any choice of prescribed subsets of vertices. A number of results on $H$-topological minor free or $H$-minor free graphs have conclusions or requirements tied to properties of $H$. Classes of graphs that can half-integrally pack only a bounded number of $H$-topological minors or $H$-minors are more general topological minor-closed or minor-closed families whose minimal obstructions are more complicated than $H$. Our theorem provides a general machinery to extend those results to those more general classes of graphs without losing their tight connections to $H$.

Figures (4)

• Figure 1: An example of the $\Delta_i$-dive $(S_i,\mu_i)$, the $\Delta_i$-disentanglement $(D_i,\sigma_i)$, and a $\Delta_i$-foundation. (1) The top side of the picture includes the graph $S_i$ and the disk $\Delta_i$, where $V(S_i)=\{v_1,v_2,...,v_{12}\}$. (2) The middle part of the picture includes the graph $D_i$ at the end of the procedure, together with the labels on the vertices of $D_i$, and the disk $\Delta_i$. The vertices in $V(D_i)-V(S_i)$ are indicated by empty circles. Each vertex in $D_i$ is labelled with a number and a word which is either "real" or "fake". In the picture, "R" denotes "real", and "F" denotes "fake". (3) The bottom side of the picture includes the graph $D_i$, a $\Delta_i$-foundation $\{\gamma_1,\gamma_2,\gamma_3,\gamma_4\}$, and disks $\Delta_i$ and $\Delta_i'$.
• Figure 2: An example for Statement \ref{['vortex_picture']} in Lemma \ref{['gauge']}. $(S,\Omega)$ is a member of ${\mathcal{S}}_2$. The cycles drawn by thin lines are the members of ${\mathcal{C}}_S$. The outer cycle drawn by a thick line is $C_S$. The inner cycle drawn by a think line is $C_S'$. Other curves and vertices denote $\pi$.
• Figure 3: The inner thick cycle is $C_S^*$. The outer thick cycle is $C_S$. The thick curves form $R_1$. The empty circles are the vertices in $Y_{S,i_{S,1}}$.
• Figure 4: A picture for vertices in parts in ${\mathcal{Q}}$. The upper side includes curves and disks related to $\bigcup_{i=1}^k\Gamma_i'$-crossings. The lower side includes the corresponding vertices in $Y^*$ and parts in ${\mathcal{Q}}$.

Theorems & Definitions (48)

• Conjecture 1.1: See k
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 4.1: rs XIII
• Lemma 4.2: l
• Lemma 4.3
• Lemma 4.4: lt