Graph Isomorphism is natural problem which is most widely believed to have intermediate complexity between $P$ and $NP$-complete. GI can be thought as deciding the existence of an isomorphism between two sets of node pairs. I'm trying to develop a notion of isomorphism between sets of triples.
We are given two hyper-graphs $G1$ and $G2$ such that the set of hyperedges ($E_1$ and $E_2$) consists of triples of 3 nodes $\{t_1, t_2, t_3 \}$. We say that Hyper-graphs $G1$ and $G2$ are isomorphic if there is bijection $f$ from $V_1$ to $V_2$ such that pair $\{u,v\} \in E1 $ if and only if $\{f(u),f(v) \} \in E2 $. We say $\{u, v \} \in E $ if there is a triple (hyperedge) $\{ u, v, z\} \in E$ for some node $z$.
Is it $NP$-complete to decide the existence of such isomorphism between two hyper-graphs $G1$ and $G2$?