The Harary reconstruction conjecture states that any graph with more than four edges can be uniquely reconstructed from its set of maximal edge-deleted subgraphs. In 1977, M\"uller verified the conjecture for graphs with $n$ vertices and $n \log_2(n)$ edges, improving on Lov\'as's bound of $\log(n^2-n)/4$. Here, we show that the reconstruction conjecture holds for graphs which have exactly one cycle and and three non-isomorphic subtrees.

翻译：Harary重构猜想指出，任何具有四条以上边的图都可以从其最大边删除子图集合中唯一重构。1977年，Müller验证了该猜想对于具有$n$个顶点和$n \log_2(n)$条边的图成立，改进了Lovás先前提出的$\log(n^2-n)/4$边界。本文证明该重构猜想对于恰好包含一个环和三个非同构子树的图同样成立。