Definition:Complement (Graph Theory)

Simple Graph
Let $$G = \left({V, E}\right)$$ be a simple graph.

The complement of $$G$$ is the simple graph $$\overline{G} = \left({V, \overline{E}}\right)$$ which consists of:
 * The same vertex set $$V$$ of $$G$$;
 * The set $$\overline{E}$$ defined such that $$\left\{{u, v}\right\} \in \overline{E} \iff \left\{{u, v}\right\} \notin E$$, where $$u$$ and $$v$$ are distinct.

Loop-Graph
If $$G = \left({V, E}\right)$$ is a loop-graph, the concept is slightly different.

The complement of $$G$$ is the simple graph $$\overline{G} = \left({V, \overline{E}}\right)$$ which consists of:
 * The same vertex set $$V$$ of $$G$$;
 * The set $$\overline{E}$$ defined such that:
 * $$\left\{{u, v}\right\} \in \overline{E} \iff \left\{{u, v}\right\} \notin E$$;
 * $$\left\{{v, v}\right\} \in \overline{E} \iff \left\{{v, v}\right\} \notin E$$.

That is, the complement $$\overline{G}$$ of a loop-graph $$G$$ has loops on all vertices where there are no loops in $$G$$.