Modular partitions and the quotient graph they define

Let $G$ be a finite simple graph (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets). A modular partition of $G$ is a set $\mathcal{P}$ of nonempty modules of $G$ (Modules of a graph, and the trivial modules) that are pairwise disjoint and whose union is $V(G)$. Its members are its parts. Since the parts are nonempty and pairwise disjoint subsets of the finite set $V(G)$, there are finitely many of them (The cardinality $\mid A\mid$ of a finite set).

The quotient graph $G\mathrm{/}\mathcal{P}$ has vertex set $\mathcal{P}$, and for distinct parts $M,N\in \mathcal{P}$,

$\{M,N\}\in E(G\mathrm{/}\mathcal{P}):⟺(M,N)\text{ is a complete pair in }G$

(Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).

Why this is a definition and not a wish. Two distinct parts are disjoint nonempty modules, so the pair they form is complete or anticomplete and not both (Two disjoint nonempty modules form a complete or an anticomplete pair). The displayed condition is therefore a genuine dichotomy: for each unordered pair of distinct parts exactly one of "complete" and "anticomplete" holds, and $E(G\mathrm{/}\mathcal{P})$ is a well-defined set of two-element subsets of $\mathcal{P}$. Hence $G\mathrm{/}\mathcal{P}$ is a finite simple graph.

The partition of $V(G)$ into singletons is modular, and its quotient is $G$ itself up to the renaming $v\mapsto \{v\}$; when $V(G)\mathrm{\ne }\mathrm{\varnothing }$ the partition $\{V(G)\}$ is modular too, and its quotient is the one-vertex graph. The induced subgraphs $G[M]$ on the parts (Subgraphs, induced subgraphs and spanning subgraphs) carry the information the quotient discards.