Definition:Complement (Graph Theory)/Loop-Graph

Definition
Let $G = \struct {V, E}$ be a loop-graph.

The complement of $G$ is the loop-graph $\overline G = \struct {V, \overline E}$ which consists of:
 * The same vertex set $V$ of $G$;
 * The set $\overline E$ defined such that:
 * $\set {u, v} \in \overline E \iff \set {u, v} \notin E$
 * $\set {v, v} \in \overline E \iff \set {v, v} \notin E$

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