Clique (graph theory)

Clique in graph theory refers to a subset of vertices within an undirected graph such that every two distinct vertices in the clique are adjacent. This means that a clique represents a set of vertices all of which are connected to each other by edges. The concept of cliques is fundamental in the study of graph theory and has applications in various fields including social network analysis, bioinformatics, and the study of computational complexity.

Definition

A clique in an undirected graph G = (V, E) is a subset C of the vertex set V such that for every two vertices in C, there exists an edge connecting them in the edge set E. The size of the largest clique in a graph is known as the graph's clique number, denoted as ω(G).

Types of Cliques

• Maximal Clique: A clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique.
• Maximum Clique: A clique of the largest possible size in a graph. This is a clique that has the highest number of vertices.

Finding Cliques

The problem of finding cliques, especially the maximum clique, in a graph is a well-known NP-complete problem, making it computationally challenging for large graphs. Various algorithms exist for finding cliques, including the Bron–Kerbosch algorithm for finding all maximal cliques and heuristic methods for finding large cliques in dense graphs.

Applications

Cliques have applications in many areas:

• In social network analysis, cliques can represent groups of people all of whom know each other.
• In bioinformatics, cliques can be used to identify groups of proteins that interact with each other.
• In computational complexity, the clique problem is used as a basis for proving the difficulty of other computational problems.

Clique Problem in Computational Complexity

The decision version of the clique problem, which asks whether a graph contains a clique of a certain size, is one of the Karp's 21 NP-complete problems. This highlights the intrinsic difficulty of solving this problem efficiently for large graphs.

See Also

• Graph theory
• NP-complete problem
• Bron–Kerbosch algorithm
• Social network analysis
• Bioinformatics