maximal_independent_set

maximal_independent_set(G, nodes=None, seed=None)

Returns a random maximal independent set guaranteed to contain a given set of nodes.

An independent set is a set of nodes such that the subgraph of G induced by these nodes contains no edges. A maximal independent set is an independent set such that it is not possible to add a new node and still get an independent set.

Parameters:

GNetworkX graph

nodeslist or iterable

Nodes that must be part of the independent set. This set of nodes must be independent.

seedinteger, random_state, or None (default)

Indicator of random number generation state. See Randomness.

Returns:

indep_nodesset

Set of nodes that are part of a maximal independent set.

Raises:

NetworkXUnfeasible

If the nodes in the provided list are not part of the graph or do not form an independent set, an exception is raised.

NetworkXNotImplemented

If G is directed.

See also

maximum_independent_set()

Algorithm for approximating the maximum independent set, i.e. a maximal independent set of maximum cardinality.

Notes

This algorithm does not solve the maximum independent set problem.

Examples

>>> G = nx.path_graph(4)
>>> sorted(nx.maximal_independent_set(G, seed=1))
[1, 3]

Note that a graph may have many maximal independent sets; for example, the path graph with 4 nodes has 3 maximal independent sets: {0, 2}, {0, 3} and {1, 3}.

>>> sorted(nx.maximal_independent_set(G, seed=2))
[0, 2]

This function returns a single maximal independent set. Enumerating all of the maximal independent sets in a graph can be achieved by enumerating the cliques in the complement graph:

>>> sorted(
...     sorted(c)
...     for c in nx.enumerate_all_cliques(nx.complement(G))
...     if len(c) > 1  # Ignore cliques comprising a single node
... )
[[0, 2], [0, 3], [1, 3]]

Note however that enumerating all cliques is exponential in the number of nodes in the worst case (e.g. the complete graph).

The nodes keyword argument can be used to produce a maximal independent set that contains the given node(s):

>>> sorted(nx.maximal_independent_set(G, nodes={1}))
[1, 3]

An exception is raised if nodes are not independent

>>> nx.maximal_independent_set(G, nodes={1, 2})
Traceback (most recent call last):
   ...
networkx.exception.NetworkXUnfeasible: {1, 2} is not an independent set of G