How do independent sets and clique partitions of a simple graph correspond to its complement? Given: Graph G and its complement $\bar{G}$
Denote C as an independent set in G.
Then, C can only be an independent set in G if it is also a clique in $\bar{G}$
Denote M as a clique partition of G.
Then, M can only be a clique partition of G if it is also a "something" in $\bar{G}$
Is the "something" a vertex set or am I missing the intuition here?
 A: Let's not call things independent sets before we've decided they're independent sets. Your first rule should read:

Let $C$ be a set of vertices of $G$ (a subset of $V(G)$. Then $C$ is an independent set in $G$ if and only if it is a clique in $\overline G$.

Of course, the relationship between complements is symmetric. So we can also say:

Let $C$ be a set of vertices of $G$. Then $C$ is a clique in $G$ if and only if it is an independent set in $\overline G$.

To understand clique partitions, it is enough to understand the rules above.
First of all, what kind of object is a clique partition? A clique partition $M$ is, first of all, a partition of the vertex set $V(G)$: it is a collection $M = \{M_1, M_2, \dots, M_k\}$ such that

*

*each $M_i \subseteq V(G)$,

*their union $M_1 \cup M_2 \cup \dots \cup M_k$ is all of $V(G)$,

*and each intersection $M_i \cap M_j$ is empty.

It is a clique partition of $G$ exactly when each $M_i$ is a clique in $G$.
Now, what happens if we take the complement? Well, $G$ and $\overline G$ have the same set of vertices: $V(G) = V(\overline G)$. So if $M$ was a partition of the vertices of $G$, then it is also a partition of the vertices of $\overline G$.
What kind of partition. Well, each $M_i$ is a clique in $G$, and we know by one of the rules above that this happens exactly when $M_i$ is an independent set in $\overline G$. So without thinking too hard, we can call the result an "independent set partition" of $\overline G$: a partition of $V(\overline G)$ into independent sets.
Now we might be able to say the following:

Let $M$ be a partition of $V(G)$. Then $M$ is a clique partition of $G$ if and only if it is an independent set partition of $\overline G$.

A final thought: think about if there is any simpler way to say "independent set partition". Is it a concept you have already encountered?
