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arXiv · 2308.04509

Acyclic graphs with at least $2\ell+1$ vertices are $\ell$-recognizable

Alexandr V. Kostochka·Mina Nahvi·Douglas B. West·Dara Zirlin
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Abstract

The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $(n-\ell)$-deck as a graph in the family is also in the family. We prove that the family of $n$-vertex graphs with no cycles is $\ell$-recognizable when $n\ge2\ell+1$ (except for $(n,\ell)=(5,2)$). As a consequence, the family of $n$-vertex trees is $\ell$-recognizable when $n\ge2\ell+1$ and $\ell\ne2$. It is known that this fails when $n=2\ell$.

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Alexandr V. Kostochka, Mina Nahvi, Douglas B. West, Dara Zirlin. 2023-08-08. Acyclic graphs with at least $2\ell+1$ vertices are $\ell$-recognizable. https://arxiv.org/abs/2308.04509

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