Maximum independent set problem

In graph theory, an independent set or stable set is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S. The size of an independent set is the number of vertices it contains. Independent sets have also been called internally stable sets.

A maximal independent set is either an independent set such that adding any other vertex to the set forces the set to contain an edge or the set of all vertices of the empty graph.

A maximum independent set is an independent set of largest possible size for a given graph G. This size is called the independence number of G, and denoted α(G). The problem of finding such a set is called the maximum independent set problem and is an NP-hard optimization problem. As such, it is unlikely that there exists an efficient algorithm for finding a maximum independent set of a graph.

Every maximum independent set also is maximal, but the converse implication does not necessarily hold.

A set is independent if and only if it is a clique in the graph’s complement, so the two concepts are complementary. In fact, sufficiently large graphs with no large cliques have large independent sets, a theme that is explored in Ramsey theory.

A set is independent if and only if its complement is a vertex cover. Therefore, the sum of the size of the largest independent set α(G), and the size of a minimum vertex cover β(G), is equal to the number of vertices in the graph.

A vertex coloring of a graph G corresponds to a partition of its vertex set into independent subsets. Hence the minimal number of colors needed in a vertex coloring, the chromatic number χ(G), is at least the quotient of the number of vertices in G and the independent number α(G).

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