Induced subdivisions with pinned branch vertices

TL;DR

This paper proves a general theorem about the structure of graphs containing certain subdivisions, showing that either large complete or bipartite induced subgraphs exist or a specific smaller subdivision with pinned branch vertices can be found.

Contribution

It establishes a new structural result for graphs with subdivisions, affirmatively answering a question by Lozin and Razgon about preserving branch vertices.

Findings

Either large complete or bipartite induced subgraphs exist

Existence of a smaller subdivision with pinned branch vertices is guaranteed

The result applies to all r, s, t with a suitable Omega

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.