# Source code for networkx.algorithms.bipartite.covering

```""" Functions related to graph covers."""

import networkx as nx
from networkx.algorithms.bipartite.matching import hopcroft_karp_matching
from networkx.algorithms.covering import min_edge_cover as _min_edge_cover
from networkx.utils import not_implemented_for

__all__ = ["min_edge_cover"]

[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable(name="bipartite_min_edge_cover")
def min_edge_cover(G, matching_algorithm=None):
"""Returns a set of edges which constitutes
the minimum edge cover of the graph.

The smallest edge cover can be found in polynomial time by finding
a maximum matching and extending it greedily so that all nodes
are covered.

Parameters
----------
G : NetworkX graph
An undirected bipartite graph.

matching_algorithm : function
A function that returns a maximum cardinality matching in a
given bipartite graph. The function must take one input, the
graph ``G``, and return a dictionary mapping each node to its
mate. If not specified,
:func:`~networkx.algorithms.bipartite.matching.hopcroft_karp_matching`
will be used. Other possibilities include
:func:`~networkx.algorithms.bipartite.matching.eppstein_matching`,

Returns
-------
set
A set of the edges in a minimum edge cover of the graph, given as
pairs of nodes. It contains both the edges `(u, v)` and `(v, u)`
for given nodes `u` and `v` among the edges of minimum edge cover.

Notes
-----
An edge cover of a graph is a set of edges such that every node of
the graph is incident to at least one edge of the set.
A minimum edge cover is an edge covering of smallest cardinality.

Due to its implementation, the worst-case running time of this algorithm
is bounded by the worst-case running time of the function
``matching_algorithm``.
"""
if G.order() == 0:  # Special case for the empty graph
return set()
if matching_algorithm is None:
matching_algorithm = hopcroft_karp_matching
return _min_edge_cover(G, matching_algorithm=matching_algorithm)

```