An independent vertex set of a graph , also known as a stable set, is a subset of the vertices such that no two vertices in the subset represent an edge of . The figure above shows independent sets consisting of two subsets for a number of graphs (the wheel graph , utility graph , Petersen graph, and Frucht graph).

Any independent vertex set is an irredundant set
(Burger *et al. *1997, Mynhardt and Roux 2020).

The polynomial whose coefficients give the number of independent vertex sets of each cardinality in a graph is known as its independence polynomial.

A set of vertices is an independent vertex set iff its complement forms a vertex cover (Skiena 1990, p. 218). The counts of vertex covers and independent vertex sets in a graph are therefore the same.

The empty set is trivially an independent vertex set since it contains no vertices, and therefore no edge endpoints.

A maximum independent vertex set is an independent vertex set of a graph containing the largest possible number of vertices for the given graph, and the cardinality of this set is called the independence number of the graph.

An independent vertex set that cannot be enlarged to another independent vertex set by adding a vertex is called a maximal independent vertex set.

In the Wolfram Language, the command `FindIndependentVertexSet`[*g*][[1]]
can be used to find a maximum independent
vertex set, and `FindIndependentVertexSet`[*g*,
`Length` /@ `FindIndependentVertexSet`[g], `All`] to find all
maximum independent vertex sets.
Similarly, `FindIndependentVertexSet`[*g*,
`Infinity`] can be used to find a maximal
independent vertex set, and `FindIndependentVertexSet`[*g*,
`Infinity`, `All`] to find all independent vertex sets. To find all
independent vertex sets in the Wolfram
Language, enumerate all vertex subsets and select those for which `IndependentVertexSetQ`[*g*,
*s*] is true.

Independent vertex set counts for some families of graphs are summarized in the following table.

graph family | OEIS | number independent vertex sets |

antiprism graph for | A000000 | X, X, 10, 21, 46, 98, 211, 453, 973, 2090, ... |

bishop graph | A201862 | X, 9, 70, 729, 9918, 167281, ... |

black bishop graph | A000000 | X, X, X, 27, 114, 409, 2066, ... |

-folded cube graph | A000000 | X, 3, 5, 31, 393, ... |

grid graph for | A006506 | X, 7, 63, 1234, 55447, 5598861, ... |

grid graph for | A000000 | X, 35, 70633, ... |

-halved cube graph | A000000 | 2, 3, 5, 13, 57, ... |

-Hanoi graph | A000000 | 4, 52, 108144, ... |

hypercube graph | A027624 | 3, 7, 35, 743, 254475, 19768832143, ... |

king graph | A063443 | X, 5, 35, 314, 6427, ... |

knight graph | A141243 | X, 16, 94, 1365, 55213, ... |

-Möbius ladder | A182143 | X, X, 15, 33, 83, 197, 479, 1153, 2787, ... |

-Mycielski graph | A000000 | 2, 3, 11, 103, 7407, ... |

odd graph | A000000 | 2, 4, 76, ... |

prism graph for | A051927 | X, X, 13, 35, 81, 199, 477, 1155, 2785, ... |

queen graph | A000000 | 2, 5, 18, 87, 462, ... |

rook graph | A002720 | 2, 7, 34, 209, 1546, 13327, 130922, ... |

-Sierpiński gasket graph | A000000 | 4, 14, 440, ... |

-triangular graph | A000000 | X, 2, 4, 10, 26, 76, 232, 764, ... |

-web graph for | A000000 | X, X, 68, 304, 1232, 5168, 21408, ... |

white bishop graph | A000000 | X, X, X, 27, 87, 409, 1657, ... |

Many families of graphs have simple closed forms for counts of independent vertex sets, as summarized in the following table. Here, is a Fibonacci number, is a Lucas number, is a Laguerre polynomial, is the golden ratio, and , , are the roots of .