Excluding $K_{2,t}$ as a fat minor

TL;DR

The paper resolves the $K_{2,t}$ case of the fat minor conjecture in coarse graph theory by proving that graphs with no $K$-fat $K_{2,t}$ minor are uniformly quasi-isometric to $K_{2,t}$-minor-free graphs, with an explicit bound $f(K)=(9t^{12}K+204t^9K,1)$, giving additive distortion $1$ and multiplicative distortion $O(K)$. This is achieved via a constructive graph-partition strategy: decompose the graph into honest, $R(K)$-bounded bags, contract to a graph $H$, and show that every 2-connected minor of $H$ yields a $K$-fat minor of the original graph using a buffer-zone construction in bags. The approach yields a polynomial-time algorithm to approximate the minimal multiplicative distortion $oldsymbol{ ho_t(G)}$ of embeddings into $K_{2,t}$-minor-free graphs, answering a 2012 question of Chepoi et al. The results advance the understanding of asymptotic minors and coarse geometry, providing a concrete, computable bridge between fat-minor conditions and standard minors, and opening avenues for similar treatments of other forbidden asymptotic minors.

Abstract

We prove that for every $t \in \mathbb{N}$, the graph $K_{2,t}$ satisfies the fat minor conjecture of Georgakopoulos and Papasoglu: for every $K\in \mathbb{N}$ there exist $M,A\in \mathbb{N}$ such that every graph with no $K$-fat $K_{2,t}$ minor is $(M,A)$-quasi-isometric to a graph with no $K_{2,t}$ minor. We use this to obtain an efficient algorithm for approximating the minimal multiplicative distortion of any embedding of a finite graph into a $K_{2,t}$-minor-free graph, answering a question of Chepoi, Dragan, Newman, Rabinovich, and Vaxès from 2012.

Figures (7)

• Figure 1: Depicted is a partition class $V_h$ of the graph-partition in Lemma \ref{['lem:GPwithNiceProperties']}. The (dark) blue vertex set $V_h^\uparrow \cup V_h^\downarrow$ is connected by \ref{['itm:FatK2t:NEW:Connected']}, and $d_G(V_h \cup V_h^\downarrow, V_{g'} \cup V^\downarrow_{g'}) \geq 3K$ holds by \ref{['itm:FatK2t:NEW:DistanceBtwBags']}.
• Figure 2: Depicted is an illustration of $U_x$, where $y,y',z \in V(J)$ and $xy, xy' \in E(J)$ and $xz \notin E(J)$.
• Figure 3: An illustration of the fat $\Theta_t$ minor in the proof of Lemma \ref{['lem:FatK2t:Components']}. The green and blue sets are its branch sets, and the orange paths are its branch paths.
• Figure 4: A visualisation of the partition $\mathfrak{R}$ of $\mathcal{C}(G-G^n)$ (in green, with dashed lines). Every partition class in $\mathfrak{R}$ is either a singleton comprising a component that has only neighbours in exactly one partition class of $\mathcal{H}^n$ (indicated in grey), or it is $\mathcal{D}_Z$ for some component $Z$ of $G-G^{n-1}$ (indicated in light/dark blue).
• Figure 5: Indicated in pink is the partition $\mathcal{B}_\mathcal{D}$. The grey boxes around the $B_i^C$ are connected, but they need not to be pairwise $3K$ far apart. Applying Lemma \ref{['lem:MergingLemma']} yields a coarsening $\mathcal{Q}_\mathcal{D}$ of $\mathcal{B}_\mathcal{D}$ such that the black boxes (of height $L_\mathcal{D}$) around the (green) partition classes of $\mathcal{Q}_\mathcal{D}$ are connected and pairwise at least $3K$ far apart. Moreover, the partition classes in $\mathcal{Q}_\mathcal{D}$ have bounded diameter.

Theorems & Definitions (24)

• Conjecture 1.1: GeoPapMin
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm:FatK2t']} given Theorem \ref{['thm:FatK2t:GraphPartition']}
• Corollary 3.2