Complement graph

File:Petersen graph complement.svg

The Petersen graph (on the left) and its complement graph (on the right).

File:Kneser graph KG(5,2).svg

The Petersen graph as Kneser graph KG(5,2) ...

File:Johnson graph J(5,2).svg

... and its complement the Johnson graph J(5,2)

In graph theory, the complement or inverse of a graph G is a graph H on the same vertices such that two distinct vertices of H are adjacent if and only if they are not adjacent in G. That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there. It is not, however, the set complement of the graph; only the edges are complemented.

Formal construction

Let G = (V, E) be a simple graph and let K consist of all 2-element subsets of V. Then H = (V, K \ E) is the complement of G.

Applications and examples

Several graph-theoretic concepts are related to each other via complement graphs:

• The vertices of the Kneser graph KG(n,k) are the k-subsets of an n-set, and the edges are between disjoint sets. The complement is the Johnson graph J(n,k), where the edges are between intersecting sets.
• The complement of an edgeless graph is a complete graph and vice versa.
• An independent set in a graph is a clique in the complement graph and vice versa.
• The complement of any triangle-free graph is a claw-free graph.
• A self-complementary graph is a graph that is isomorphic to its own complement.
• Cographs are defined as the graphs that can be built up from disjoint union and complementation operations, and form a self-complementary family of graphs: the complement of any cograph is another (possibly different) cograph.

References

• , page 6.
• . Electronic edition, page 4.