Fixed database typo and removed unnecessary class identifier.

venv/Lib/site-packages/networkx/algorithms/covering.py

@@ -0,0 +1,115 @@
+""" Functions related to graph covers."""
+
+import networkx as nx
+from networkx.utils import not_implemented_for, arbitrary_element
+from functools import partial
+from itertools import chain
+
+
+__all__ = ["min_edge_cover", "is_edge_cover"]
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+def min_edge_cover(G, matching_algorithm=None):
+    """Returns a set of edges which constitutes
+    the minimum edge cover of the graph.
+
+    A 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`,
+        or matching algorithms in the
+        :mod:`networkx.algorithms.matching` module.
+
+    Returns
+    -------
+    min_cover : set
+
+        It contains all the edges of minimum edge cover
+        in form of tuples. 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.
+    The 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``.
+
+    Minimum edge cover for bipartite graph can also be found using the
+    function present in :mod:`networkx.algorithms.bipartite.covering`
+    """
+    if nx.number_of_isolates(G) > 0:
+        # ``min_cover`` does not exist as there is an isolated node
+        raise nx.NetworkXException(
+            "Graph has a node with no edge incident on it, " "so no edge cover exists."
+        )
+    if matching_algorithm is None:
+        matching_algorithm = partial(nx.max_weight_matching, maxcardinality=True)
+    maximum_matching = matching_algorithm(G)
+    # ``min_cover`` is superset of ``maximum_matching``
+    try:
+        min_cover = set(
+            maximum_matching.items()
+        )  # bipartite matching case returns dict
+    except AttributeError:
+        min_cover = maximum_matching
+    # iterate for uncovered nodes
+    uncovered_nodes = set(G) - {v for u, v in min_cover} - {u for u, v in min_cover}
+    for v in uncovered_nodes:
+        # Since `v` is uncovered, each edge incident to `v` will join it
+        # with a covered node (otherwise, if there were an edge joining
+        # uncovered nodes `u` and `v`, the maximum matching algorithm
+        # would have found it), so we can choose an arbitrary edge
+        # incident to `v`. (This applies only in a simple graph, not a
+        # multigraph.)
+        u = arbitrary_element(G[v])
+        min_cover.add((u, v))
+        min_cover.add((v, u))
+    return min_cover
+
+
+@not_implemented_for("directed")
+def is_edge_cover(G, cover):
+    """Decides whether a set of edges is a valid edge cover of the graph.
+
+    Given a set of edges, whether it is an edge covering can
+    be decided if we just check whether all nodes of the graph
+    has an edge from the set, incident on it.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        An undirected bipartite graph.
+
+    cover : set
+        Set of edges to be checked.
+
+    Returns
+    -------
+    bool
+        Whether the set of edges is a valid edge cover of the graph.
+
+    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.
+    """
+    return set(G) <= set(chain.from_iterable(cover))