diff --git a/doc/reference/algorithms/planarity.rst b/doc/reference/algorithms/planarity.rst
index cad00dc8e..03ded5fa8 100644
--- a/doc/reference/algorithms/planarity.rst
+++ b/doc/reference/algorithms/planarity.rst
@@ -7,5 +7,5 @@ Planarity
    :toctree: generated/
 
    check_planarity
-.. autoclass:: PlanarEmbedding
-   :members:
\ No newline at end of file
+   is_planar
+   PlanarEmbedding
diff --git a/networkx/algorithms/bridges.py b/networkx/algorithms/bridges.py
index 959a67fec..eda4a4f6d 100644
--- a/networkx/algorithms/bridges.py
+++ b/networkx/algorithms/bridges.py
@@ -35,6 +35,9 @@ def bridges(G, root=None):
     NodeNotFound
        If `root` is not in the graph `G`.
 
+    NetworkXNotImplemented
+        If `G` is a directed graph.
+
     Examples
     --------
     The barbell graph with parameter zero has a single bridge:
@@ -99,6 +102,9 @@ def has_bridges(G, root=None):
     NodeNotFound
        If `root` is not in the graph `G`.
 
+    NetworkXNotImplemented
+        If `G` is a directed graph.
+
     Examples
     --------
     The barbell graph with parameter zero has a single bridge::
@@ -158,6 +164,11 @@ def local_bridges(G, with_span=True, weight=None):
         The local bridges as an edge 2-tuple of nodes `(u, v)` or
         as a 3-tuple `(u, v, span)` when `with_span is True`.
 
+    Raises
+    ------
+    NetworkXNotImplemented
+        If `G` is a directed graph or multigraph.
+
     Examples
     --------
     A cycle graph has every edge a local bridge with span N-1.
diff --git a/networkx/algorithms/community/modularity_max.py b/networkx/algorithms/community/modularity_max.py
index 264a0dc07..ca1cfb9e2 100644
--- a/networkx/algorithms/community/modularity_max.py
+++ b/networkx/algorithms/community/modularity_max.py
@@ -326,6 +326,8 @@ def greedy_modularity_communities(
             raise ValueError(f"best_n must be between 1 and {len(G)}. Got {best_n}.")
         if best_n < cutoff:
             raise ValueError(f"Must have best_n >= cutoff. Got {best_n} < {cutoff}")
+        if best_n == 1:
+            return [set(G)]
     else:
         best_n = G.number_of_nodes()
     if n_communities is not None:
@@ -351,7 +353,19 @@ def greedy_modularity_communities(
 
     # continue merging communities until one of the breaking criteria is satisfied
     while len(communities) > cutoff:
-        dq = next(community_gen)
+        try:
+            dq = next(community_gen)
+        # StopIteration occurs when communities are the connected components
+        except StopIteration:
+            communities = sorted(communities, key=len, reverse=True)
+            # if best_n requires more merging, merge big sets for highest modularity
+            while len(communities) > best_n:
+                comm1, comm2, *rest = communities
+                communities = [comm1 ^ comm2]
+                communities.extend(rest)
+            return communities
+
+        # keep going unless max_mod is reached or best_n says to merge more
         if dq < 0 and len(communities) <= best_n:
             break
         communities = next(community_gen)
diff --git a/networkx/algorithms/covering.py b/networkx/algorithms/covering.py
index 174044270..b2aeb82b5 100644
--- a/networkx/algorithms/covering.py
+++ b/networkx/algorithms/covering.py
@@ -12,36 +12,42 @@ __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.
+    """Returns the min cardinality edge cover of the graph as a set of edges.
 
     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.
+    are covered. This function follows that process. A maximum matching
+    algorithm can be specified for the first step of the algorithm.
+    The resulting set may return a set with one 2-tuple for each edge,
+    (the usual case) or with both 2-tuples `(u, v)` and `(v, u)` for
+    each edge. The latter is only done when a bipartite matching algorithm
+    is specified as `matching_algorithm`.
 
     Parameters
     ----------
     G : NetworkX graph
-        An undirected bipartite graph.
+        An undirected 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,
+        A function that returns a maximum cardinality matching for `G`.
+        The function must take one input, the graph `G`, and return
+        either a set of edges (with only one direction for the pair of nodes)
+        or a dictionary mapping each node to its mate. If not specified,
+        :func:`~networkx.algorithms.matching.max_weight_matching` is used.
+        Common bipartite matching functions include
         :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.
+        or
+        :func:`~networkx.algorithms.bipartite.matching.eppstein_matching`.
 
     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.
+        A set of the edges in a minimum edge cover in the form of tuples.
+        It contains only one of the equivalent 2-tuples `(u, v)` and `(v, u)`
+        for each edge. If a bipartite method is used to compute the matching,
+        the returned set contains both the 2-tuples `(u, v)` and `(v, u)`
+        for each edge of a minimum edge cover.
 
     Notes
     -----
@@ -53,9 +59,13 @@ def min_edge_cover(G, matching_algorithm=None):
     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`
+    Minimum edge cover for `G` can also be found using the `min_edge_covering`
+    function in :mod:`networkx.algorithms.bipartite.covering` which is
+    simply this function with a default matching algorithm of
+    :func:`~networkx.algorithms.bipartite.matching.hopcraft_karp_matching`
     """
+    if len(G) == 0:
+        return set()
     if nx.number_of_isolates(G) > 0:
         # ``min_cover`` does not exist as there is an isolated node
         raise nx.NetworkXException(
@@ -66,11 +76,12 @@ def min_edge_cover(G, matching_algorithm=None):
     maximum_matching = matching_algorithm(G)
     # ``min_cover`` is superset of ``maximum_matching``
     try:
-        min_cover = set(
-            maximum_matching.items()
-        )  # bipartite matching case returns dict
+        # bipartite matching algs return dict so convert if needed
+        min_cover = set(maximum_matching.items())
+        bipartite_cover = True
     except AttributeError:
         min_cover = maximum_matching
+        bipartite_cover = False
     # 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:
@@ -82,7 +93,8 @@ def min_edge_cover(G, matching_algorithm=None):
         # multigraph.)
         u = arbitrary_element(G[v])
         min_cover.add((u, v))
-        min_cover.add((v, u))
+        if bipartite_cover:
+            min_cover.add((v, u))
     return min_cover
 
 
diff --git a/networkx/algorithms/planarity.py b/networkx/algorithms/planarity.py
index 4d1441efc..bc6ff2f0f 100644
--- a/networkx/algorithms/planarity.py
+++ b/networkx/algorithms/planarity.py
@@ -1,7 +1,39 @@
 from collections import defaultdict
 import networkx as nx
 
-__all__ = ["check_planarity", "PlanarEmbedding"]
+__all__ = ["check_planarity", "is_planar", "PlanarEmbedding"]
+
+
+def is_planar(G):
+    """Returns True if and only if `G` is planar.
+
+    A graph is *planar* iff it can be drawn in a plane without
+    any edge intersections.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    Returns
+    -------
+    bool
+       Whether the graph is planar.
+
+    Examples
+    --------
+    >>> G = nx.Graph([(0, 1), (0, 2)])
+    >>> nx.is_planar(G)
+    True
+    >>> nx.is_planar(nx.complete_graph(5))
+    False
+
+    See Also
+    --------
+    check_planarity :
+        Check if graph is planar *and* return a `PlanarEmbedding` instance if True.
+    """
+
+    return check_planarity(G, counterexample=False)[0]
 
 
 def check_planarity(G, counterexample=False):
@@ -24,6 +56,18 @@ def check_planarity(G, counterexample=False):
         If the graph is planar `certificate` is a PlanarEmbedding
         otherwise it is a Kuratowski subgraph.
 
+    Examples
+    --------
+    >>> G = nx.Graph([(0, 1), (0, 2)])
+    >>> is_planar, P = nx.check_planarity(G)
+    >>> print(is_planar)
+    True
+
+    When `G` is planar, a `PlanarEmbedding` instance is returned:
+
+    >>> P.get_data()
+    {0: [1, 2], 1: [0], 2: [0]}
+
     Notes
     -----
     A (combinatorial) embedding consists of cyclic orderings of the incident
@@ -37,6 +81,11 @@ def check_planarity(G, counterexample=False):
     A counterexample is only generated if the corresponding parameter is set,
     because the complexity of the counterexample generation is higher.
 
+    See also
+    --------
+    is_planar :
+        Check for planarity without creating a `PlanarEmbedding` or counterexample.
+
     References
     ----------
     .. [1] Ulrik Brandes:
@@ -716,6 +765,8 @@ class PlanarEmbedding(nx.DiGraph):
     The planar embedding is given by a `combinatorial embedding
     <https://en.wikipedia.org/wiki/Graph_embedding#Combinatorial_embedding>`_.
 
+    .. note:: `check_planarity` is the preferred way to check if a graph is planar.
+
     **Neighbor ordering:**
 
     In comparison to a usual graph structure, the embedding also stores the
@@ -761,6 +812,15 @@ class PlanarEmbedding(nx.DiGraph):
     For a half-edge (u, v) that is orientated such that u is below v then the
     face that belongs to (u, v) is to the right of this half-edge.
 
+    See Also
+    --------
+    is_planar :
+        Preferred way to check if an existing graph is planar.
+
+    check_planarity :
+        A convenient way to create a `PlanarEmbedding`. If not planar,
+        it returns a subgraph that shows this.
+
     Examples
     --------
 
diff --git a/networkx/algorithms/tournament.py b/networkx/algorithms/tournament.py
index bace64bae..a002e022e 100644
--- a/networkx/algorithms/tournament.py
+++ b/networkx/algorithms/tournament.py
@@ -82,6 +82,13 @@ def is_tournament(G):
     bool
         Whether the given graph is a tournament graph.
 
+    Examples
+    --------
+    >>> from networkx.algorithms import tournament
+    >>> G = nx.DiGraph([(0, 1), (1, 2), (2, 0)])
+    >>> tournament.is_tournament(G)
+    True
+
     Notes
     -----
     Some definitions require a self-loop on each node, but that is not
@@ -114,6 +121,13 @@ def hamiltonian_path(G):
     path : list
         A list of nodes which form a Hamiltonian path in `G`.
 
+    Examples
+    --------
+    >>> from networkx.algorithms import tournament
+    >>> G = nx.DiGraph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
+    >>> tournament.hamiltonian_path(G)
+    [0, 1, 2, 3]
+
     Notes
     -----
     This is a recursive implementation with an asymptotic running time
@@ -185,6 +199,13 @@ def score_sequence(G):
     list
         A sorted list of the out-degrees of the nodes of `G`.
 
+    Examples
+    --------
+    >>> from networkx.algorithms import tournament
+    >>> G = nx.DiGraph([(1, 0), (1, 3), (0, 2), (0, 3), (2, 1), (3, 2)])
+    >>> tournament.score_sequence(G)
+    [1, 1, 2, 2]
+
     """
     return sorted(d for v, d in G.out_degree())
 
@@ -260,6 +281,15 @@ def is_reachable(G, s, t):
     bool
         Whether there is a path from `s` to `t` in `G`.
 
+    Examples
+    --------
+    >>> from networkx.algorithms import tournament
+    >>> G = nx.DiGraph([(1, 0), (1, 3), (1, 2), (2, 3), (2, 0), (3, 0)])
+    >>> tournament.is_reachable(G, 1, 3)
+    True
+    >>> tournament.is_reachable(G, 3, 2)
+    False
+
     Notes
     -----
     Although this function is more theoretically efficient than the
@@ -331,6 +361,16 @@ def is_strongly_connected(G):
     bool
         Whether the tournament is strongly connected.
 
+    Examples
+    --------
+    >>> from networkx.algorithms import tournament
+    >>> G = nx.DiGraph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3), (3, 0)])
+    >>> tournament.is_strongly_connected(G)
+    True
+    >>> G.remove_edge(1, 3)
+    >>> tournament.is_strongly_connected(G)
+    False
+
     Notes
     -----
     Although this function is more theoretically efficient than the
diff --git a/networkx/algorithms/tree/recognition.py b/networkx/algorithms/tree/recognition.py
index 52da959b6..88b2abe8b 100644
--- a/networkx/algorithms/tree/recognition.py
+++ b/networkx/algorithms/tree/recognition.py
@@ -96,6 +96,16 @@ def is_arborescence(G):
     b : bool
         A boolean that is True if `G` is an arborescence.
 
+    Examples
+    --------
+    >>> G = nx.DiGraph([(0, 1), (0, 2), (2, 3), (3, 4)])
+    >>> nx.is_arborescence(G)
+    True
+    >>> G.remove_edge(0, 1)
+    >>> G.add_edge(1, 2)  # maximum in-degree is 2
+    >>> nx.is_arborescence(G)
+    False
+
     Notes
     -----
     In another convention, an arborescence is known as a *tree*.
@@ -125,6 +135,16 @@ def is_branching(G):
     b : bool
         A boolean that is True if `G` is a branching.
 
+    Examples
+    --------
+    >>> G = nx.DiGraph([(0, 1), (1, 2), (2, 3), (3, 4)])
+    >>> nx.is_branching(G)
+    True
+    >>> G.remove_edge(2, 3)
+    >>> G.add_edge(3, 1)  # maximum in-degree is 2
+    >>> nx.is_branching(G)
+    False
+
     Notes
     -----
     In another convention, a branching is also known as a *forest*.

diff --git a/networkx/algorithms/community/tests/test_modularity_max.py b/networkx/algorithms/community/tests/test_modularity_max.py
index 784e4a97b..acdb19d64 100644
--- a/networkx/algorithms/community/tests/test_modularity_max.py
+++ b/networkx/algorithms/community/tests/test_modularity_max.py
@@ -44,6 +44,17 @@ def test_modularity_communities_categorical_labels(func):
     assert set(func(G)) == expected
 
 
+def test_greedy_modularity_communities_components():
+    # Test for gh-5530
+    G = nx.Graph([(0, 1), (2, 3), (4, 5), (5, 6)])
+    # usual case with 3 components
+    assert greedy_modularity_communities(G) == [{4, 5, 6}, {0, 1}, {2, 3}]
+    # best_n can make the algorithm continue even when modularity goes down
+    assert greedy_modularity_communities(G, best_n=3) == [{4, 5, 6}, {0, 1}, {2, 3}]
+    assert greedy_modularity_communities(G, best_n=2) == [{0, 1, 4, 5, 6}, {2, 3}]
+    assert greedy_modularity_communities(G, best_n=1) == [{0, 1, 2, 3, 4, 5, 6}]
+
+
 def test_greedy_modularity_communities_relabeled():
     # Test for gh-4966
     G = nx.balanced_tree(2, 2)
@@ -306,7 +317,7 @@ def test_cutoff_parameter():
 def test_best_n():
     G = nx.barbell_graph(5, 3)
 
-    # Same result as without enforcing n_communities:
+    # Same result as without enforcing cutoff:
     best_n = 3
     expected = [frozenset(range(5)), frozenset(range(8, 13)), frozenset(range(5, 8))]
     assert greedy_modularity_communities(G, best_n=best_n) == expected
diff --git a/networkx/algorithms/tests/test_covering.py b/networkx/algorithms/tests/test_covering.py
index 78487b734..40971963e 100644
--- a/networkx/algorithms/tests/test_covering.py
+++ b/networkx/algorithms/tests/test_covering.py
@@ -14,28 +14,44 @@ class TestMinEdgeCover:
         assert nx.min_edge_cover(G) == {(0, 0)}
 
     def test_graph_single_edge(self):
-        G = nx.Graph()
-        G.add_edge(0, 1)
+        G = nx.Graph([(0, 1)])
         assert nx.min_edge_cover(G) in ({(0, 1)}, {(1, 0)})
 
+    def test_graph_two_edge_path(self):
+        G = nx.path_graph(3)
+        min_cover = nx.min_edge_cover(G)
+        assert len(min_cover) == 2
+        for u, v in G.edges:
+            assert (u, v) in min_cover or (v, u) in min_cover
+
     def test_bipartite_explicit(self):
         G = nx.Graph()
         G.add_nodes_from([1, 2, 3, 4], bipartite=0)
         G.add_nodes_from(["a", "b", "c"], bipartite=1)
         G.add_edges_from([(1, "a"), (1, "b"), (2, "b"), (2, "c"), (3, "c"), (4, "a")])
+        # Use bipartite method by prescribing the algorithm
         min_cover = nx.min_edge_cover(
             G, nx.algorithms.bipartite.matching.eppstein_matching
         )
-        min_cover2 = nx.min_edge_cover(G)
         assert nx.is_edge_cover(G, min_cover)
         assert len(min_cover) == 8
+        # Use the default method which is not specialized for bipartite
+        min_cover2 = nx.min_edge_cover(G)
+        assert nx.is_edge_cover(G, min_cover2)
+        assert len(min_cover2) == 4
 
-    def test_complete_graph(self):
+    def test_complete_graph_even(self):
         G = nx.complete_graph(10)
         min_cover = nx.min_edge_cover(G)
         assert nx.is_edge_cover(G, min_cover)
         assert len(min_cover) == 5
 
+    def test_complete_graph_odd(self):
+        G = nx.complete_graph(11)
+        min_cover = nx.min_edge_cover(G)
+        assert nx.is_edge_cover(G, min_cover)
+        assert len(min_cover) == 6
+
 
 class TestIsEdgeCover:
     """Tests for :func:`networkx.algorithms.is_edge_cover`"""

greedy_modularity_communities raises StopIteration for unconnected (?) graph
<!--- Provide a general summary of the issue in the Title above -->

After an update of `networkx` from 2.6.3 to 2.8, one of my tests suddenly failed with
```
Traceback (most recent call last):
  File "/usr/lib/python3.10/runpy.py", line 196, in _run_module_as_main
    return _run_code(code, main_globals, None,
  File "/usr/lib/python3.10/runpy.py", line 86, in _run_code
    exec(code, run_globals)
  File "/home/gereon/Develop/lexedata/src/lexedata/report/nonconcatenative_morphemes.py", line 159, in <module>
    networkx.algorithms.community.greedy_modularity_communities(graph),
  File "/home/gereon/.local/etc/lexedata/lib/python3.10/site-packages/networkx/algorithms/community/modularity_max.py", line 354, in greedy_modularity_communities
    dq = next(community_gen)
StopIteration

```
It turns out that the internals of `greedy_modularity_communities` have changed, so the function doesn't work any more with unconnected graphs.

### Current Behavior
<!--- Tell us what happens instead of the expected behavior -->

`networkx.algorithms.community.greedy_modularity_communities` fails with undocumented error message for unconnected graphs, or graphs without edges

### Expected Behavior
<!--- Tell us what should happen -->

 - Either document the requirement to have a connected graph with at least one edge and throw `ValueError` instead of `StopIteration` when it is not fulfilled, or run the graph through `nx.connected_components()` first.
 - If the graph has only one node and no edges, return that node.

### Steps to Reproduce
<!--- Provide a minimal example that reproduces the bug -->

```
>>> G = nx.Graph()
>>> G.add_edges_from([(0,1),(2,3)])
>>> greedy_modularity_communities(G)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/home/gereon/.local/etc/lexedata/lib/python3.10/site-packages/networkx/algorithms/community/modularity_max.py", line 354, in greedy_modularity_communities
    dq = next(community_gen)
StopIteration
```

```
>>> G = nx.Graph()
>>> G.add_nodes_from([1])
>>> greedy_modularity_communities(G)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/home/gereon/.local/etc/lexedata/lib/python3.10/site-packages/networkx/algorithms/community/modularity_max.py", line 350, in greedy_modularity_communities
    communities = next(community_gen)
  File "/home/gereon/.local/etc/lexedata/lib/python3.10/site-packages/networkx/algorithms/community/modularity_max.py", line 79, in _greedy_modularity_communities_generator
    q0 = 1 / m
ZeroDivisionError: division by zero

```
### Environment
<!--- Please provide details about your local environment -->
Python version: 3.10.2
NetworkX version: 2.8
networkx/algorithms/community/tests/test_modularity_max.py::test_greedy_modularity_communities_components networkx/algorithms/tests/test_covering.py::TestMinEdgeCover::test_graph_two_edge_path networkx/algorithms/tests/test_covering.py::TestMinEdgeCover::test_bipartite_explicit networkx/algorithms/tests/test_covering.py::TestMinEdgeCover::test_complete_graph_odd
networkx/algorithms/community/tests/test_modularity_max.py::test_modularity_communities[greedy_modularity_communities] networkx/algorithms/community/tests/test_modularity_max.py::test_modularity_communities[naive_greedy_modularity_communities] networkx/algorithms/community/tests/test_modularity_max.py::test_modularity_communities_categorical_labels[greedy_modularity_communities] networkx/algorithms/community/tests/test_modularity_max.py::test_modularity_communities_categorical_labels[naive_greedy_modularity_communities] networkx/algorithms/community/tests/test_modularity_max.py::test_greedy_modularity_communities_relabeled networkx/algorithms/community/tests/test_modularity_max.py::test_greedy_modularity_communities_directed networkx/algorithms/community/tests/test_modularity_max.py::test_modularity_communities_weighted[greedy_modularity_communities] networkx/algorithms/community/tests/test_modularity_max.py::test_modularity_communities_weighted[naive_greedy_modularity_communities] networkx/algorithms/community/tests/test_modularity_max.py::test_modularity_communities_floating_point networkx/algorithms/community/tests/test_modularity_max.py::test_modularity_communities_directed_weighted networkx/algorithms/community/tests/test_modularity_max.py::test_greedy_modularity_communities_multigraph networkx/algorithms/community/tests/test_modularity_max.py::test_greedy_modularity_communities_multigraph_weighted networkx/algorithms/community/tests/test_modularity_max.py::test_greed_modularity_communities_multidigraph networkx/algorithms/community/tests/test_modularity_max.py::test_greed_modularity_communities_multidigraph_weighted networkx/algorithms/community/tests/test_modularity_max.py::test_resolution_parameter_impact networkx/algorithms/community/tests/test_modularity_max.py::test_cutoff_parameter networkx/algorithms/community/tests/test_modularity_max.py::test_best_n networkx/algorithms/tests/test_covering.py::TestMinEdgeCover::test_empty_graph networkx/algorithms/tests/test_covering.py::TestMinEdgeCover::test_graph_with_loop networkx/algorithms/tests/test_covering.py::TestMinEdgeCover::test_graph_single_edge networkx/algorithms/tests/test_covering.py::TestMinEdgeCover::test_complete_graph_even networkx/algorithms/tests/test_covering.py::TestIsEdgeCover::test_empty_graph networkx/algorithms/tests/test_covering.py::TestIsEdgeCover::test_graph_with_loop networkx/algorithms/tests/test_covering.py::TestIsEdgeCover::test_graph_single_edge