Independent set (graph theory)

In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set ${\displaystyle S}$ of vertices such that for every two vertices in ${\displaystyle S}$, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in ${\displaystyle S}$. A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening.^[1]

A maximal independent set is an independent set that is not a proper subset of any other independent set.

A maximum independent set is an independent set of largest possible size for a given graph ${\displaystyle G}$. This size is called the independence number of ${\displaystyle G}$ and is usually denoted by ${\displaystyle \alpha (G)}$.^[2] The optimization problem of finding such a set is called the maximum independent set problem. It is a strongly NP-hard problem.^[3] 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.