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arXiv · 2308.01502

Induced subdivisions with pinned branch vertices

Sepehr Hajebi
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Abstract

We prove that for all $r\in \mathbb{N}\cup \{0\}$ and $s,t\in \mathbb{N}$, there exists $\Omega=\Omega(r,s,t)\in \mathbb{N}$ with the following property. Let $G$ be a graph and let $H$ be a subgraph of $G$ isomorphic to a $(\leq r)$-subdivision of $K_{\Omega}$. Then either $G$ contains $K_t$ or $K_{t,t}$ as an induced subgraph, or there is an induced subgraph $J$ of $G$ isomorphic to a proper $(\leq r)$-subdivision of $K_s$ such that every branch vertex of $J$ is a branch vertex of $H$. This answers in the affirmative a question of Lozin and Razgon. In fact, we show that both the branch vertices and the paths corresponding to the subdivided edges between them can be preserved.

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Sepehr Hajebi. 2023-08-03. Induced subdivisions with pinned branch vertices. https://doi.org/10.1016/j.ejc.2024.104072

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