Polynomial Bounds in the Apex Minor Theorem

TL;DR

This work addresses the problem of bounding the treewidth of $A$-minor-free graphs when $A$ is apex and the radius is bounded by $r$. It introduces a simple proof of the Apex Minor Theorem and, more importantly, establishes a polynomial bound $g(A,r) in O^*(r^9 t^{18})$ with $t=|V(A)|$, by showing the absence of large grid minors $16rt^2 × 16rt^2$ and applying the Polynomial Grid Minor Theorem. It further refines the bounds for the case $A=K_{3,t}$, proving no $n imes n$ grid minor exists for $n=ig ceil 4r(1+ ext{√}(t-1)) ig ceil$, yielding $ ext{tw}(G) in O^*(r^9 t^{9/2})$ (tight up to constants) and extending these ideas to graphs embeddable on fixed surfaces. The paper also connects these bounds to the recently studied tree-treewidth, deriving explicit polynomial bounds in terms of $t=|V(A)|$, and presents lower bounds showing the near-tightness of the results, along with several open problems including tighter grid-minor bounds and a conjectural bound on a related contraction-parameter ell$(p,q)$. These results advance the understanding of how apex-minor-freeness interacts with radius to control treewidth without relying on the heavy Graph Minor Structure Theorem.

Abstract

A graph $A$ is "apex" if $A-z$ is planar for some vertex $z\in V(A)$. Eppstein [Algorithmica, 2000] showed that for a minor-closed class $\mathcal{G}$, the graphs in $\mathcal{G}$ with bounded radius have bounded treewidth if and only if some apex graph is not in $\mathcal{G}$. In particular, for every apex graph $A$ and integer $r$, there is a minimum integer $g(A,r)$ such that every $A$-minor-free graph with radius $r$ has treewidth at most $g(A,r)$. We show that if $t=|V(A)|$ then $g(A,r)\in O^\ast(r^9t^{18})$ which is the first upper bound on $g(A,r)$ with polynomial dependence on both $r$ and $t$. More precisely, we show that every $A$-minor-free graph with radius $r$ has no $16rt^2 \times 16rt^2$ grid minor, which implies the first result via the Polynomial Grid Minor Theorem. A key example of an apex graph is the complete bipartite graph $K_{3,t}$, since $K_{3,t}$-minor-free graphs include and generalise graphs embeddable in any fixed surface. In this case, we prove that every $K_{3,t}$-minor-free graph with radius $r$ has no $4r(1+\sqrt{t})\times 4r(1+\sqrt{t})$ grid minor, which is tight up to a constant factor.

Figures (3)

• Figure 1: Example with $H=K_4$: (left) the model $\{B_u:u\in V(H)\}$ in a grid, highlighting the vertex $h_u$, (right) the model $\{C_u:u\in V(H)\}$ in a larger grid, where dashed edges are bars arising from neighbouring blocks.
• Figure 2: Proof of \ref{['K3tGridRadius']}: the subgraphs $X'$ and $Y'$ are shown in red and blue; the vertical paths in $X'\cup Y'$ are chosen randomly from within each shaded block; and the vertices $x\in A"'$ are green, enlarged into the paths $Z_x$.
• Figure 3: Construction in the proof of \ref{['LowerBound']} with $r=k=3$, where vertices in $W$ are yellow.

Theorems & Definitions (29)

• Theorem 1: RS-V
• Lemma 2: RST94
• Theorem 3: RS-VRST94
• Theorem 4: Eppstein-Algo00
• Theorem 5
• Corollary 6
• proof
• Theorem 7
• Corollary 8
• Proposition 9