Harary's edge reconstruction conjecture

Let $G=(V,E)$) be a finite simple graph with $|E|>4$. For each edge of $G$, let $G_i$ denote the maximal subgraph obtained by deleting that edge, and define the edge deck of $G$ by

$$\operatorname{Deck}(G)\coloneqq\{G_i\}_{i=1}^m.$$

Harary's reconstruction conjecture. If $\operatorname{Deck}(G)=\operatorname{Deck}(H)$ for some graph $H$, then $G\cong H$.

This conjecture asserts that every graph with more than four edges is uniquely determined, up to isomorphism, by its maximal edge-deleted subgraphs. The paper proves it for graphs with exactly one cycle and three non-isomorphic subtrees; the general conjecture remains open.