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arXiv · 2608.14196

Product structure of graphs excluding a topological minor

Jędrzej Hodor·Hoang La·Piotr Micek·Clément Rambaud
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Abstract

We prove that, for all positive integers $h$ and $t$ and every graph $X$ with $\mathrm{td}(X) \leq h$, there exists a positive integer $c(X,t)$ such that every graph $G$ with $\mathrm{tw}(G) < t$ that excludes $X$ as a topological minor is isomorphic to a subgraph of $H \boxtimes K_{c(X,t)}$ for some graph $H$ with $\mathrm{tw}(H) < 2^{h+1}-1$. This extends a result by Ding and Oporowski (Journal of Graph Theory; 1995), which states that for all positive integers $\Delta$ and $t$, here exists a positive integer $f(\Delta,t)$ such that every graph $G$ with $\mathrm{tw}(G)<t$ and $\Delta(G)\leq\Delta$ is isomorphic to a subgraph of $T \boxtimes K_{f(\Delta,t)}$ for some tree $T$.

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Jędrzej Hodor, Hoang La, Piotr Micek, Clément Rambaud. 2026-08-14. Product structure of graphs excluding a topological minor. https://arxiv.org/abs/2608.14196

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