从更小的卡片重构
Reconstruction from smaller cards
AI总结：
本文研究图的重构问题，证明树可从更小的$(n-r)$-牌组重构，其中$r\le n/9+o(n)$，并改进连通性识别与度序列重构的界限，显著推进Nýdl猜想。
摘要：
图$G$的$\ell$-牌组是$G$的所有$\ell$个顶点上的诱导子图的多重集。我们说一个图可以从它的$\ell$-牌组重构，如果没有其他图具有相同的$\ell$-牌组。1957年，Kelly证明了每个具有$n\ge3$个顶点的树都可以从其$(n-1)$-牌组重构，Giles在1976年加强了这一点，证明了至少有6个顶点的树可以从它们的$(n-2)$-牌组重构。我们的主要定理表明，对于所有$r\le n/{9}+o(n)$，树可以从它们的$(n-r)$-牌组重构，这朝着Nýdl在1990年提出的猜想取得了实质性进展。此外，当$\ell\ge 9n/10$时，我们可以从图的$\ell$-牌组识别其连通性，并且当$\ell\ge\sqrt{2n\log(2n)}$时，可以重构度序列。所有这些结果都是对先前界限的显著改进。
英文摘要：
The $\ell$-deck of a graph $G$ is the multiset of all induced subgraphs of $G$ on $\ell$ vertices. We say that a graph is reconstructible from its $\ell$-deck if no other graph has the same $\ell$-deck. In 1957, Kelly showed that every tree with $n\ge3$ vertices can be reconstructed from its $(n-1)$-deck, and Giles strengthened this in 1976, proving that trees on at least 6 vertices can be reconstructed from their $(n-2)$-decks. Our main theorem states that trees are reconstructible from their $(n-r)$-decks for all $r\le n/{9}+o(n)$, making substantial progress towards a conjecture of Nýdl from 1990. In addition, we can recognise the connectedness of a graph from its $\ell$-deck when $\ell\ge 9n/10$, and reconstruct the degree sequence when $\ell\ge\sqrt{2n\log(2n)}$. All of these results are significant improvements on previous bounds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
