Tagged graph families with anticomplete parts

Let $T$ be a finite set and let $J_t=(V(J_t),E(J_t))$ be a finite simple graph for each $t\in T$. The disjoint graph family of the $J_t$ is the finite simple graph $D=(V(D),E(D))$ with vertex set $V(D)={\bigcup }_{t\in T}(\{t\}\times V(J_t))$ and edge set $E(D)=\{\{(t,v),(t,w)\}:t\in T,\{v,w\}\in E(J_t)\}$. The set $\{t\}\times V(J_t)$ is the tagged part for $t$. Distinct tagged parts are anticomplete: no edge of $D$ has one endpoint in each. The construction permits an empty index set or empty parts.

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

The Cartesian product A × B := { z ∈ P(P(A ∪ B)) : ∃ a ∈ A ∃ b ∈ B z = (a,b) }

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

For each $t\in T$, the singleton $\{t\}$ is finite because $\{t\}\subseteq T$ and a subset of a finite set is finite. Each tagged part is therefore finite by the finite product rule. Distinct tagged parts are disjoint because equality of ordered pairs implies equality of their first coordinates. The finite sum rule therefore makes their union $V(D)$ finite. Each specified edge is a two-element subset of $V(D)$: its endpoints are distinct because $\{v,w\}$ is an edge of the simple graph $J_t$. Thus $D$ is a finite simple graph. Its edge set is specified uniquely, has no edges between distinct parts, and within the part tagged $t$ the map $v\mapsto (t,v)$ preserves and reflects adjacency. If $T$ is empty, both vertex and edge sets are empty.

The product rule: | A × B| = | A| | B|, and |∏_i<m Aᵢ| = ∏_i<m| Aᵢ|

The sum rule: a finite disjoint union is finite with | A ∪ B| = | A| + | B| and |⋃_i ∈ I Aᵢ| = ∑_i ∈ I| Aᵢ|, and a sum over a finite index set splits along a partition

(a,b) = (c,d) if and only if a = c and b = d

A subset of a finite set is finite, with | B| ≤ | A|, and equality holds if and only if B = A

Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree