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تسجيل مستخدم جديدShotgun assembly of unlabeled Erdos-Renyi graphs
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Given an unlabeled graph $G$ on $n$ vertices, let ${N_{G}(v)}_{v}$ be the collection of subgraphs of $G$, where for each vertex $v$ of $G$, $N_{G}(v)$ is the subgraph of $G$ induced by vertices of $G$ of distance at most one from $v$. We show that there are universal constants $C,c>0$ with the following property. Let the sequence $(p_n)_{n=1}^infty$ satisfy $n^{-1/2}log^C nleq p_nleq c$. For each $n$, let $Gamma_n$ be an unlabeled $G(n,p_n)$ Erdos-Renyi graph. Then with probability $1-o(1)$, any unlabeled graph $tilde Gamma_n$ on $n$ vertices with ${N_{tilde Gamma_n}(v)}_{v}={N_{Gamma_n}(v)}_{v}$ must coincide with $Gamma_n$. This establishes $p_n= tilde Theta(n^{-1/2})$ as the transition for the density parameter $p_n$ between reconstructability and non-reconstructability of Erdos-Renyi graphs from their $1$--neighborhoods, answering a question of Gaudio and Mossel.
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