图重构猜想的一个代数表述
An algebraic formulation of the graph reconstruction conjecture
摘要：
图重构猜想断言，每个至少有三个顶点的有限简单图都可以从它的甲板（它的顶点删除子图的集合）重构到同构。Kocay引理是图重构中的一个重要工具。粗略地说，给定图$G$和任何有限图序列的甲板，它给出了$G$的每个重建必须满足的线性约束。令$ψ(n)$是$n$顶点上不同（相互不同构）图的数量，令$d(n)$是可以从这些图构造的不同甲板的数量。然后，差异$ψ(n) - d(n)$衡量有多少图形无法从其套牌中重建。特别地，图重构猜想对$n$ -顶点图成立当且仅当$ψ(n) = d(n)$ .我们给出了一个框架的基础上Kocay的引理来研究这种差异。证明了若$M$是图的序列覆盖数矩阵，则$d(n) \geq \mathsf{rank}_\mathbb{R}(M)$ .特别地，如果一个这样的矩阵具有秩$ψ(n)$，则所有$n$ -顶点图都是可重构的.为了补充这一结果，我们证明了可以选择一族图序列，使得对应的覆盖数矩阵$M$满足$d(n) = \mathsf{rank}_\mathbb{R}(M)$。
英文摘要：
The graph reconstruction conjecture asserts that every finite simple graph on at least three vertices can be reconstructed up to isomorphism from its deck - the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important tool in graph reconstruction. Roughly speaking, given the deck of a graph $G$ and any finite sequence of graphs, it gives a linear constraint that every reconstruction of $G$ must satisfy. Let $ψ(n)$ be the number of distinct (mutually non-isomorphic) graphs on $n$ vertices, and let $d(n)$ be the number of distinct decks that can be constructed from these graphs. Then the difference $ψ(n) - d(n)$ measures how many graphs cannot be reconstructed from their decks. In particular, the graph reconstruction conjecture is true for $n$-vertex graphs if and only if $ψ(n) = d(n)$. We give a framework based on Kocay's lemma to study this discrepancy. We prove that if $M$ is a matrix of covering numbers of graphs by sequences of graphs, then $d(n) \geq \mathsf{rank}_\mathbb{R}(M)$. In particular, all $n$-vertex graphs are reconstructible if one such matrix has rank $ψ(n)$. To complement this result, we prove that it is possible to choose a family of sequences of graphs such that the corresponding matrix $M$ of covering numbers satisfies $d(n) = \mathsf{rank}_\mathbb{R}(M)$.
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