Graph isomorphism conjectures for edge-, spanning-tree-, and subset-deleted graphs

Let $G,H,G_a,G_b$ be graphs, let $E_G\subset E(G)$ and $E_H\subset E(H)$ be edge subsets, let $T_a,T_b$ be spanning trees, and let $S_G\in V(G)\cup E(G)$ and $S_H\in V(H)\cup E(H)$ be proper subsets. Graph isomorphism conjectures. (i) If $|E_G|=|E_H|$, $G-E_G\cong H-E_H$, and $G,H$ are connected $(p,q)$-graphs admitting the isomorphic subgraph similarity, then $G\cong H$ by Kelly–Ulam's Reconstruction Conjecture. (ii) If each spanning tree $T_a$ of a connected $(p,q)$-graph $G_a$ corresponds to a spanning tree $T_b$ of another connected graph $G_b$ with $T_a\cong T_b$, and vice versa, then $G_a\cong G_b$. (iii) If $G$ and $H$ have $n$ vertices and each proper subset $S_G\in V(G)\cup E(G)$ corresponds to a proper subset $S_H\in V(H)\cup E(H)$ with $G-S_G\cong H-S_H$, then $G\cong H$. These proposed extensions of reconstruction from deleted subgraphs concern when graph isomorphism is determined by families of edge-, spanning-tree-, or subset-deleted graphs; the source gives no resolution status.