Category:Complements of Graphs

This category contains results about Complements of Graphs.
Definitions specific to this category can be found in Definitions/Complements of Graphs.

Simple Graph

Let $G = \struct {V, E}$ be a simple graph.

The complement of $G$ is the simple graph $\overline G = \struct {V, \overline E}$ which consists of:

The same vertex set $V$ of $G$

The set $\overline E$ defined such that $\set {u, v} \in \overline E \iff \set {u, v} \notin E$, where $u$ and $v$ are distinct.

Loop-Graph

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 $G$.