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arXiv · 2607.03405

Oddomorphisms, Split-Off Minors, and the Strong Roberson Conjecture

Arnar Á. Kristjánsson
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Abstract

We show that the existence of an oddomorphism from a graph $F$ to a graph $G$ does not imply that $G$ is a minor of $F$. This answers a question posed by Roberson (2022) and shows that the CFI graphs cannot be used to prove the Strong Roberson Conjecture. Additionally, we introduce the concept of a split-off minor and show that the existence of an oddomorphism from $F$ to $G$ implies that $G$ is a split-off minor of $F$. Consequently, every class that is closed under taking split-off minors and disjoint unions is homomorphism distinguishing closed. The split-off minor relation is the first minor-like structural relation shown to have this property, marking a meaningful advancement in our understanding of the interaction between structural graph containment and homomorphism indistinguishability relations.

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Arnar Á. Kristjánsson. 2026-07-03. Oddomorphisms, Split-Off Minors, and the Strong Roberson Conjecture. https://arxiv.org/abs/2607.03405

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