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On Cliques in Graphs
The number of cliques in a graph: I added the BibTeX for reference: John Moeller 2 The actual tight lower bound is by removing a set of disjoint triangles instead of a perfect matching. Thank you very much for taking the time to write out a very detailed answer. Suresh Venkat 27k 4 79 The answers that have been given so far are great.
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I thought I'd add some references. The Moon-Moser theorem was independently proved by Miller and Muller  in a technical report. Wood  and Vatter  give simpler proofs of the Theorem, using basically the approach outlined by David.
Serge Gaspers 4, 23 A related concept is a biclique , a complete bipartite subgraph. The bipartite dimension of a graph is the minimum number of bicliques needed to cover all the edges of the graph.
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Closely related concepts to complete subgraphs are subdivisions of complete graphs and complete graph minors. In particular, Kuratowski's theorem and Wagner's theorem characterize planar graphs by forbidden complete and complete bipartite subdivisions and minors, respectively.
In computer science , the clique problem is the computational problem of finding a maximum clique, or all cliques, in a given graph. Nevertheless, many algorithms for computing cliques have been developed, either running in exponential time such as the Bron—Kerbosch algorithm or specialized to graph families such as planar graphs or perfect graphs for which the problem can be solved in polynomial time. For continued efforts to model social cliques graph-theoretically, see e.
Many different problems from bioinformatics have been modeled using cliques. Sugihara uses cliques to model ecological niches in food webs. Power graph analysis is a method for simplifying complex biological networks by finding cliques and related structures in these networks. Cliques have also been used in automatic test pattern generation: In automatic test pattern generation , finding cliques can help to bound the size of a test set.
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.
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A clique in a graph G is a complete subgraph of G. That is, it is a subset K of the vertices such that every two vertices in K are the two endpoints of an edge in G. A maximal clique is a clique to which no more vertices can be added. For each vertex v that is not part of a maximal clique, there must be another vertex w that is in the clique and non-adjacent to v , preventing v from being added to the clique.
A maximum clique is a clique that includes the largest possible number of vertices. Several closely related clique-finding problems have been studied. The first four of these problems are all important in practical applications. The clique decision problem is not of practical importance; it is formulated in this way in order to apply the theory of NP-completeness to clique-finding problems.
The clique problem and the independent set problem are complementary: 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. A maximal clique, sometimes called inclusion-maximal, is a clique that is not included in a larger clique. Therefore, every clique is contained in a maximal clique.
Clique (graph theory) - Wikipedia
A graph may contain a non-maximal clique with many vertices and a separate clique of size 2 which is maximal. While a maximum i. There are some types of graphs in which every maximal clique is maximum; these are the complements of the well-covered graphs , in which every maximal independent set is maximum. A single maximal clique can be found by a straightforward greedy algorithm. Starting with an arbitrary clique for instance, any single vertex or even the empty set , grow the current clique one vertex at a time by looping through the graph's remaining vertices.
For each vertex v that this loop examines, add v to the clique if it is adjacent to every vertex that is already in the clique, and discard v otherwise.
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This algorithm runs in linear time. However, some research in parallel algorithms has studied the problem of finding a maximal clique. In particular, the problem of finding the lexicographically first maximal clique the one found by the algorithm above has been shown to be complete for the class of polynomial-time functions. This result implies that the problem is unlikely to be solvable within the parallel complexity class NC. One can test whether a graph G contains a k -vertex clique, and find any such clique that it contains, using a brute force algorithm.
This algorithm examines each subgraph with k vertices and checks to see whether it forms a clique. This is because there are O n k subgraphs to check, each of which has O k 2 edges whose presence in G needs to be checked. Thus, the problem may be solved in polynomial time whenever k is a fixed constant.
However, when k does not have a fixed value, but instead may vary as part of the input to the problem, the time is exponential. The simplest nontrivial case of the clique-finding problem is finding a triangle in a graph, or equivalently determining whether the graph is triangle-free. The worst case for this formula occurs when G is itself a clique. To do so the algorithm marks all neighbors of v , searches through all edges incident to a neighbor of v outputting a triangle for every edge that has two marked endpoints, and then removes the marks and deletes v from the graph.
For graphs of constant arboricity, such as planar graphs or in general graphs from any non-trivial minor-closed graph family , this algorithm takes O m time, which is optimal since it is linear in the size of the input. If one desires only a single triangle, or an assurance that the graph is triangle-free, faster algorithms are possible.
Therefore, fast matrix multiplication techniques such as the Coppersmith—Winograd algorithm can be applied to find triangles in time O n 2. These algorithms based on fast matrix multiplication have also been extended to problems of finding k -cliques for larger values of k. The main recursive subroutine of this procedure has three arguments: The algorithm tries adding the candidate vertices one by one to the partial clique, making a recursive call for each one. After trying each of these vertices, it moves it to the set of vertices that should not be added again.
Further, the Bron—Kerbosch algorithm has been widely reported as being faster in practice than its alternatives. However, when the number of cliques is significantly smaller than its worst case, other algorithms might be preferable. As Tsukiyama et al. An algorithm such as theirs in which the running time depends on the output size is known as an output-sensitive algorithm. On the basis of this principle, they show that all maximal cliques in G may be generated in time O mn per clique, where m is the number of edges in G and n is the number of vertices.
However, the choice of ordering is important for the efficiency of this algorithm: On the basis of this result, it is possible to list all maximal cliques in polynomial time, for families of graphs in which the number of cliques is polynomially bounded. These families include chordal graphs , complete graphs , triangle-free graphs , interval graphs , graphs of bounded boxicity , and planar graphs.
The same is true for any family of graphs that is both sparse having a number of edges at most a constant times the number of vertices and closed under the operation of taking subgraphs. However, for this variant of the clique problem better worst-case time bounds are possible. It is a recursive backtracking scheme similar to that of the Bron—Kerbosch algorithm , but is able to eliminate some recursive calls when it can be shown that the cliques found within the call will be suboptimal. Jian improved the time to O 2 0. Robson's algorithm combines a similar backtracking scheme with a more complicated case analysis and a dynamic programming technique in which the optimal solution is precomputed for all small connected subgraphs of the complement graph.
These partial solutions are used to shortcut the backtracking recursion. The fastest algorithm known today is a refined version of this method by Robson which runs in time O 2 0. There has also been extensive research on heuristic algorithms for solving maximum clique problems without worst-case runtime guarantees, based on methods including branch and bound ,  local search ,  greedy algorithms ,  and constraint programming.
Planar graphs , and other families of sparse graphs, have been discussed above: