Modules of a graph, and the trivial modules

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 vertex set $M\subseteq V(G)$ is a module of $G$ when every vertex $v\in V(G)\setminus M$ is adjacent to every vertex of $M$ or to no vertex of $M$. Equivalently, the disjoint pair $(\{v\},M)$ is pure for every $v\in V(G)\setminus M$ (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs, Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

The condition constrains only the edges between $M$ and $V(G)\setminus M$: no condition whatever is placed on the induced subgraph $G[M]$ (Subgraphs, induced subgraphs and spanning subgraphs).

The trivial modules of $G$ are $\mathrm{\varnothing }$, the singletons $\{v\}$ for $v\in V(G)$, and $V(G)$ itself. Each of the three really is a module: for $M=\mathrm{\varnothing }$ every pair $(\{v\},\mathrm{\varnothing })$ is both complete and anticomplete, hence pure; for $M=\{u\}$ the pair $(\{v\},\{u\})$ is complete when $uv\in E(G)$ and anticomplete otherwise; and for $M=V(G)$ there is no vertex outside $M$, so the condition is vacuous. A module that is not one of these is nontrivial. Since $V(G)$ is finite, a module $M$ is nontrivial exactly when $2\le \mathrm{\mid }M\mathrm{\mid }$ and $\mathrm{\mid }M\mathrm{\mid }\le \mathrm{\mid }V(G)\mathrm{\mid }-1$, the second bound because a subset of a finite set has the full cardinality only if it is the whole set (The cardinality $\mid A\mid$ of a finite set, A subset of a finite set is finite, with $\mid B\mid \le \mid A\mid$, and equality holds if and only if $B=A$).

A module $M$ is proper when $M\mathrm{\ne }V(G)$. Thus $\mathrm{\varnothing }$ is a proper module exactly when $V(G)\mathrm{\ne }\mathrm{\varnothing }$, every singleton of a graph with at least two vertices is a proper module, and every nontrivial module is proper.

The word module is Habib and Paul's. The same object is called a clan by Harju, a closed set by Gallai, and an autonomous, partitive, externally related or homogeneous set elsewhere; the clash between the last of these and the published meaning of homogeneous set is the subject of Why this page says module where some sources say homogeneous set.