Graph realisations with possibly different hyperedge pieces

A finite hypergraph here is a pair $(V,\mathcal{E})$ where $V$ is a finite set and $\mathcal{E}$ is a set of subsets of $V$; its members are called hyperedges. For each $e\in \mathcal{E}$, let $J_e=(e,E_e)$ be a finite simple graph with vertex set exactly $e$. The hyperedge-indexed graph realisation of these specified pieces is the finite simple graph $R=(V,E_R)$, where $E_R={\bigcup }_{e\in \mathcal{E}}{E}_e$. Each $J_e$ is called the piece indexed by $e$.

If a piece is specified by a template graph $T_e=(W_e,F_e)$ and a bijection $f_e:W_e\to e$, its transported edge set is $E_e=\{\{f_e(x),f_e(y)\}:\{x,y\}\in F_e\}$. The specified bijection is part of this construction. The induced graph $R[e]$ has vertex set $e$ and consists of all edges of $R$ with both endpoints in $e$; the definition does not assert that $R[e]=J_e$.

A finite simple graph is a finite vertex set together with a set of two-element vertex subsets

Injection, surjection, bijection

The union ⋃ x of a set, and the binary union a ∪ b := ⋃ {a,b}

An indexed family (Aᵢ)_i ∈ I is a function with domain I; {Aᵢ : i ∈ I} is its range

⋃_i ∈ I Aᵢ := ⋃ {Aᵢ : i ∈ I}, and ⋂_i ∈ I Aᵢ := ⋂ {Aᵢ : i ∈ I} for I ≠ ∅

Each edge in $E_e$ is a two-element subset of $e$, hence of $V$. The union therefore contains only two-element subsets of $V$, and so defines a finite simple graph on $V$; if $\mathcal{E}$ is empty, the union is empty. For a transported piece, injectivity of $f_e$ preserves the distinctness of the endpoints and its codomain places them in $e$. The formula uniquely specifies its edge set, and the union uniquely specifies $E_R$. Edges contributed by other pieces may have both endpoints in $e$, so equality with the induced graph is a separate assertion.