Definitions
An undirected graph is formed by a finite set of vertices and a set of unordered pairs of vertices, which are called edges. By convention, in algorithm analysis, the number of vertices in the graph is denoted by n and the number of edges is denoted by m. A clique in a graph G is a complete subgraph of G; that is, it is a subset S of the vertices such that every two vertices in S form an edge in G. A maximal clique is a clique to which no more vertices can be added; a maximum clique is a clique that includes the largest possible number of vertices, and the clique number ω(G) is the number of vertices in a maximum clique of G.
Several closely related clique-finding problems have been studied.
- In the maximum clique problem, the input is an undirected graph, and the output is a maximum clique in the graph. If there are multiple maximum cliques, only one need be output.
- In the weighted maximum clique problem, the input is an undirected graph with weights on its vertices (or, less frequently, edges) and the output is a clique with maximum total weight. The maximum clique problem is the special case in which all weights are one.
- In the maximal clique listing problem, the input is an undirected graph, and the output is a list of all its maximal cliques. The maximum clique problem may be solved using as a subroutine an algorithm for the maximal clique listing problem, because the maximum clique must be included among all the maximal cliques.
- In the k-clique problem, the input is an undirected graph and a number k, and the output is a clique of size k if one exists (or, sometimes, all cliques of size k).
- In the clique decision problem, the input is an undirected graph and a number k, and the output is a Boolean value: true if the graph contains a k-clique, and false otherwise.
The first four of these problems are all important in practical applications; the clique decision problem is not, but is necessary in order to apply the theory of NP-completeness to clique-finding problems.
The clique problem and the independent set problem are complementary: a clique in G is an independent set in the complement graph of G and vice versa. Therefore, many computational results may be applied equally well to either problem, and some research papers do not clearly distinguish between the two problems. However, the two problems have different properties when applied to restricted families of graphs; for instance, the clique problem may be solved in polynomial time for planar graphs while the independent set problem remains NP-hard on planar graphs.
Read more about this topic: Clique Problem
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