# Source code for networkx.algorithms.dominating

```"""Functions for computing dominating sets in a graph."""
from itertools import chain

import networkx as nx
from networkx.utils import arbitrary_element

__all__ = ["dominating_set", "is_dominating_set"]

[docs]def dominating_set(G, start_with=None):
r"""Finds a dominating set for the graph G.

A *dominating set* for a graph with node set *V* is a subset *D* of
*V* such that every node not in *D* is adjacent to at least one
member of *D* [1]_.

Parameters
----------
G : NetworkX graph

start_with : node (default=None)
Node to use as a starting point for the algorithm.

Returns
-------
D : set
A dominating set for G.

Notes
-----
This function is an implementation of algorithm 7 in [2]_ which
finds some dominating set, not necessarily the smallest one.

--------
is_dominating_set

References
----------
.. [1] https://en.wikipedia.org/wiki/Dominating_set

.. [2] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf

"""
all_nodes = set(G)
if start_with is None:
start_with = arbitrary_element(all_nodes)
if start_with not in G:
raise nx.NetworkXError(f"node {start_with} is not in G")
dominating_set = {start_with}
dominated_nodes = set(G[start_with])
remaining_nodes = all_nodes - dominated_nodes - dominating_set
while remaining_nodes:
# Choose an arbitrary node and determine its undominated neighbors.
v = remaining_nodes.pop()
undominated_neighbors = set(G[v]) - dominating_set
# Add the node to the dominating set and the neighbors to the
# dominated set. Finally, remove all of those nodes from the set
# of remaining nodes.
dominated_nodes |= undominated_neighbors
remaining_nodes -= undominated_neighbors
return dominating_set

[docs]@nx._dispatch
def is_dominating_set(G, nbunch):
"""Checks if `nbunch` is a dominating set for `G`.

A *dominating set* for a graph with node set *V* is a subset *D* of
*V* such that every node not in *D* is adjacent to at least one
member of *D* [1]_.

Parameters
----------
G : NetworkX graph

nbunch : iterable
An iterable of nodes in the graph `G`.