complement graph

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

The complement graph of a complete graph is an empty graph.

Equal to vertex coloring in the complement graph.

A 'graph antihole' is the complement graph of a graph hole.

These subgraphs are all isomorphic to the complement graph of the Clebsch graph.

The complement graph of every comparability graph is also a string graph.

An independent set in a graph is a clique in the complement graph and vice versa.

Their complement graph are comparability graphs, and the comparability relations are precisely the interval orders.

Analogously, Dilworth's theorem states that every complement graph of a comparability graph is perfect.

Ramsey's theorem states that every graph or its complement graph contains a clique with at least a logarithmic number of vertices.

Connected components of a graph , or of its complement graph are also modules of .

The simplex graph of the complement graph of a path graph is a Fibonacci cube.

It may be viewed as the clique complex of the complement graph of the line graph of the given graph.

The opposite of a clique is an independent set, in the sense that every clique corresponds to an independent set in the complement graph.

In the complement graphs of bipartite graphs, König's theorem allows the maximum clique problem to be solved using techniques for matching.

Additionally, he proved that, with a single exception (the eight-vertex complement graph of the cube) every nearly planar graph has an embedding onto the projective plane.

The n-vertex edgeless graph is the complement graph for the complete graph , and therefore it is commonly denoted as .

These same families of graphs also show up in connections between the Colin de Verdière invariant of a graph and the structure of its complement graph:

In graph theory, the perfect graph theorem of states that an undirected graph is perfect if and only if its complement graph is also perfect.

The complement graph of a 2n-vertex crown graph is the Cartesian product of complete graphs K K, or equivalently the 2 x n rook's graph.

Another way of constructing the Moser spindle is as the complement graph of the graph formed from the utility graph K by subdividing one of its edges.

Claw-free graphs are the graphs that are locally co-triangle-free; that is, for all vertices, the complement graph of the neighborhood of the vertex does not contain a triangle.

The complement graph of a critical graph is necessarily matching-critical, a fact that was used by Gallai to prove lower bounds on the number of vertices in a critical graph.

Somewhat more efficiently, but more complicatedly, one can test whether a graph is claw-free by checking, for each vertex of the graph, that the complement graph of its neighbors does not contain a triangle.

The Hajnal-Szemerédi theorem may be recovered from this conjecture by applying the conjecture for larger values of k to the complement graph of a given graph, and using as color classes contiguous subsequences of vertices from the n-cycle.