Graph Theory Questions Long
In graph theory, a clique is a subset of vertices in an undirected graph where every pair of vertices is connected by an edge. In other words, a clique is a complete subgraph within a larger graph.
Formally, a clique in a graph G is a subset C of vertices such that for every pair of vertices u and v in C, there exists an edge between u and v. This means that every vertex in C is directly connected to every other vertex in C.
The size of a clique is defined as the number of vertices it contains. A clique of size 2 is simply an edge, while a clique of size 3 is a triangle, and so on. The maximum clique in a graph is the largest clique that can be found within the graph.
Clique is an important concept in graph theory as it helps in understanding the connectivity and structure of a graph. It provides insights into the relationships and interactions between vertices. Cliques can be used to model various real-world scenarios, such as social networks, where a clique represents a group of individuals who are all connected to each other.
Finding cliques in a graph is a computationally challenging problem, known as the clique problem. It is an NP-complete problem, meaning that there is no known efficient algorithm to solve it in polynomial time. However, there are several algorithms and heuristics that can be used to find cliques in practice, such as the Bron-Kerbosch algorithm and the clique percolation method.
In summary, a clique in a graph is a subset of vertices where every pair of vertices is connected by an edge. It provides insights into the connectivity and structure of a graph and can be used to model various real-world scenarios. Finding cliques in a graph is a challenging problem, but there are algorithms and heuristics available to tackle it.