What are cliques in a graph?
A clique, , in an undirected graph is a subset of the vertices, , such that every two distinct vertices are adjacent. This is equivalent to the condition that the induced subgraph of induced by. is a complete graph. In some cases, the term clique may also refer to the subgraph directly.
Are cliques complete graphs?
A complete graph is often called a clique. The size of the largest clique that can be made up of edges and vertices of G is called the clique number of G.
How do you check if a graph is a clique?
To find a clique of G:
- Suppose that G has n vertices.
- Find a vertex v of the smallest possible degree in G.
- If the degree of v is n − 1, stop; G is a clique, so the largest clique in G has size n.
- Otherwise, remove v and all of its edges from G. Find the largest clique in the smaller graph.
How many cliques are in a complete graph?
from each other). 0-cliques correspond to the empty set (sets of 0 vertices), 1-cliques correspond to vertices, 2-cliques to edges, and 3-cliques to 3-cycles. , etc….Clique.
| graph family | OEIS | number of cliques |
|---|---|---|
| complete bipartite graph | A000290 | 4, 9, 16, 25, 36, 49, 64, 81, 100. |
Is clique a NP?
The clique decision problem is NP-complete (one of Karp’s 21 NP-complete problems). The problem of finding the maximum clique is both fixed-parameter intractable and hard to approximate.
How many edges does a clique graph have?
There are 5 cliques on one vertex (the five vertices) and 6 cliques on two vertices (the six edges).
Why clique problem is NP?
The clique decision problem is NP-complete (one of Karp’s 21 NP-complete problems). The problem of finding the maximum clique is both fixed-parameter intractable and hard to approximate. And, listing all maximal cliques may require exponential time as there exist graphs with exponentially many maximal cliques.
Can a clique be a single vertex?
Yes. We can have a clique of size 1 (the vertex itself) which comprises the entire graph and is connected to every other vertex (itself).