gh-36232: Make `min_spanning_tree` robust to incomparable vertex labels
    
Part of #35902.

### 📚 Description

We ensure that method `min_spanning_tree` operates properly even when
vertices and edges are of incomparable types.
We are then able to simplify some code in
`src/sage/topology/simplicial_complex.py`.

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- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
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### ⌛ Dependencies

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URL: #36232
Reported by: David Coudert
Reviewer(s): Frédéric Chapoton

Author: Release Manager

src/sage/graphs/base/boost_graph.pyx

@@ -686,6 +686,16 @@ cpdef min_spanning_tree(g,
         Traceback (most recent call last):
         ...
         TypeError: float() argument must be a string or a... number...
+
+    Check that the method is robust to incomparable vertices::
+
+        sage: G = Graph([(1, 2, 10), (1, 'a', 1), ('a', 'b', 1), ('b', 2, 1)], weighted=True)
+        sage: E = min_spanning_tree(G, algorithm='Kruskal')
+        sage: sum(w for _, _, w in E)
+        3
+        sage: F = min_spanning_tree(G, algorithm='Prim')
+        sage: sum(w for _, _, w in F)
+        3
     """
     from sage.graphs.graph import Graph
 
@@ -719,9 +729,8 @@ cpdef min_spanning_tree(g,
 
     if <v_index> result.size() != 2 * (n - 1):
         return []
-    else:
-        edges = [(int_to_vertex[<int> result[2*i]], int_to_vertex[<int> result[2*i + 1]]) for i in range(n - 1)]
-        return [(min(e[0], e[1]), max(e[0], e[1]), g.edge_label(e[0], e[1])) for e in edges]
+    edges = [(int_to_vertex[<int> result[2*i]], int_to_vertex[<int> result[2*i + 1]]) for i in range(n - 1)]
+    return [(u, v, g.edge_label(u, v)) for u, v in edges]
 
 
 cpdef blocks_and_cut_vertices(g):

src/sage/graphs/generic_graph.py

@@ -4704,17 +4704,23 @@ def min_spanning_tree(self,
             sage: len(g.min_spanning_tree())
             4
             sage: weight = lambda e: 1 / ((e[0] + 1) * (e[1] + 1))
-            sage: sorted(g.min_spanning_tree(weight_function=weight))
-            [(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]
-            sage: sorted(g.min_spanning_tree(weight_function=weight,
-            ....:                            algorithm='Kruskal_Boost'))
-            [(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]
+            sage: E = g.min_spanning_tree(weight_function=weight)
+            sage: T = Graph(E)
+            sage: set(g) == set(T) and T.order() == T.size() + 1 and T.is_tree()
+            True
+            sage: sum(map(weight, E))
+            5/12
+            sage: E = g.min_spanning_tree(weight_function=weight,
+            ....:                         algorithm='Kruskal_Boost')
+            sage: Graph(E).is_tree(); sum(map(weight, E))
+            True
+            5/12
             sage: g = graphs.PetersenGraph()
             sage: g.allow_multiple_edges(True)
             sage: g.add_edges(g.edge_iterator())
-            sage: sorted(g.min_spanning_tree())
-            [(0, 1, None), (0, 4, None), (0, 5, None), (1, 2, None), (1, 6, None),
-             (3, 8, None), (5, 7, None), (5, 8, None), (6, 9, None)]
+            sage: T = Graph(g.min_spanning_tree())
+            sage: set(g) == set(T) and T.order() == T.size() + 1 and T.is_tree()
+            True
 
         Boruvka's algorithm::
 
@@ -4725,15 +4731,13 @@ def min_spanning_tree(self,
         Prim's algorithm::
 
             sage: g = graphs.CompleteGraph(5)
-            sage: sorted(g.min_spanning_tree(algorithm='Prim_edge',
-            ....:                            starting_vertex=2, weight_function=weight))
-            [(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]
-            sage: sorted(g.min_spanning_tree(algorithm='Prim_fringe',
-            ....:                            starting_vertex=2, weight_function=weight))
-            [(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]
-            sage: sorted(g.min_spanning_tree(weight_function=weight,
-            ....:                            algorithm='Prim_Boost'))
-            [(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]
+            sage: for algo in ['Prim_edge', 'Prim_fringe', 'Prim_Boost']:
+            ....:     E = g.min_spanning_tree(algorithm=algo, weight_function=weight)
+            ....:     T = Graph(E)
+            ....:     print(set(g) == set(T) and T.order() == T.size() + 1 and T.is_tree())
+            True
+            True
+            True
 
         NetworkX algorithm::
 
@@ -4745,82 +4749,93 @@ def min_spanning_tree(self,
             sage: G = Graph([(0, 1, {'name': 'a', 'weight': 1}),
             ....:            (0, 2, {'name': 'b', 'weight': 3}),
             ....:            (1, 2, {'name': 'b', 'weight': 1})])
-            sage: sorted(G.min_spanning_tree(weight_function=lambda e: e[2]['weight']))
+            sage: sorted(G.min_spanning_tree(algorithm='Boruvka',
+            ....:                            weight_function=lambda e: e[2]['weight']))
             [(0, 1, {'name': 'a', 'weight': 1}), (1, 2, {'name': 'b', 'weight': 1})]
 
         If the graph is not weighted, edge labels are not considered, even if
         they are numbers::
 
             sage: g = Graph([(1, 2, 1), (1, 3, 2), (2, 3, 1)])
-            sage: sorted(g.min_spanning_tree())
+            sage: sorted(g.min_spanning_tree(algorithm='Boruvka'))
             [(1, 2, 1), (1, 3, 2)]
 
         In order to use weights, we need either to set variable ``weighted`` to
         ``True``, or to specify a weight function or set by_weight to ``True``::
 
             sage: g.weighted(True)
-            sage: sorted(g.min_spanning_tree())
+            sage: Graph(g.min_spanning_tree()).edges(sort=True)
             [(1, 2, 1), (2, 3, 1)]
             sage: g.weighted(False)
-            sage: sorted(g.min_spanning_tree())
+            sage: Graph(g.min_spanning_tree()).edges(sort=True)
             [(1, 2, 1), (1, 3, 2)]
-            sage: sorted(g.min_spanning_tree(by_weight=True))
+            sage: Graph(g.min_spanning_tree(by_weight=True)).edges(sort=True)
+            [(1, 2, 1), (2, 3, 1)]
+            sage: Graph(g.min_spanning_tree(weight_function=lambda e: e[2])).edges(sort=True)
             [(1, 2, 1), (2, 3, 1)]
-            sage: sorted(g.min_spanning_tree(weight_function=lambda e: e[2]))
+
+        Note that the order of the vertices on each edge is not guaranteed and
+        may differ from an algorithm to the other::
+
+            sage: g.weighted(True)
+            sage: sorted(g.min_spanning_tree())
+            [(2, 1, 1), (3, 2, 1)]
+            sage: sorted(g.min_spanning_tree(algorithm='Boruvka'))
+            [(1, 2, 1), (2, 3, 1)]
+            sage: Graph(g.min_spanning_tree()).edges(sort=True)
             [(1, 2, 1), (2, 3, 1)]
 
+
         TESTS:
 
         Check that, if ``weight_function`` is not provided, then edge weights
         are used::
 
             sage: g = Graph(weighted=True)
             sage: g.add_edges([[0, 1, 1], [1, 2, 1], [2, 0, 10]])
-            sage: sorted(g.min_spanning_tree())
+            sage: Graph(g.min_spanning_tree()).edges(sort=True)
             [(0, 1, 1), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Filter_Kruskal'))
+            sage: Graph(g.min_spanning_tree(algorithm='Filter_Kruskal')).edges(sort=True)
             [(0, 1, 1), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Kruskal_Boost'))
+            sage: Graph(g.min_spanning_tree(algorithm='Kruskal_Boost')).edges(sort=True)
             [(0, 1, 1), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Prim_fringe'))
+            sage: Graph(g.min_spanning_tree(algorithm='Prim_fringe')).edges(sort=True)
             [(0, 1, 1), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Prim_edge'))
+            sage: Graph(g.min_spanning_tree(algorithm='Prim_edge')).edges(sort=True)
             [(0, 1, 1), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Prim_Boost'))
+            sage: Graph(g.min_spanning_tree(algorithm='Prim_Boost')).edges(sort=True)
             [(0, 1, 1), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='NetworkX'))                     # needs networkx
+            sage: Graph(g.min_spanning_tree(algorithm='Boruvka')).edges(sort=True)
             [(0, 1, 1), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Boruvka'))
+            sage: Graph(g.min_spanning_tree(algorithm='NetworkX')).edges(sort=True)     # needs networkx
             [(0, 1, 1), (1, 2, 1)]
 
         Check that, if ``weight_function`` is provided, it overrides edge
         weights::
 
             sage: g = Graph([[0, 1, 1], [1, 2, 1], [2, 0, 10]], weighted=True)
             sage: weight = lambda e: 3 - e[0] - e[1]
-            sage: sorted(g.min_spanning_tree(weight_function=weight))
+            sage: Graph(g.min_spanning_tree(weight_function=weight)).edges(sort=True)
             [(0, 2, 10), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Filter_Kruskal', weight_function=weight))
+            sage: Graph(g.min_spanning_tree(algorithm='Filter_Kruskal', weight_function=weight)).edges(sort=True)
             [(0, 2, 10), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Kruskal_Boost', weight_function=weight))
+            sage: Graph(g.min_spanning_tree(algorithm='Kruskal_Boost', weight_function=weight)).edges(sort=True)
             [(0, 2, 10), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Prim_fringe', weight_function=weight))
+            sage: Graph(g.min_spanning_tree(algorithm='Prim_fringe', weight_function=weight)).edges(sort=True)
             [(0, 2, 10), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Prim_edge', weight_function=weight))
+            sage: Graph(g.min_spanning_tree(algorithm='Prim_edge', weight_function=weight)).edges(sort=True)
             [(0, 2, 10), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Prim_Boost', weight_function=weight))
+            sage: Graph(g.min_spanning_tree(algorithm='Prim_Boost', weight_function=weight)).edges(sort=True)
             [(0, 2, 10), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='NetworkX', weight_function=weight))         # needs networkx
-            [(0, 2, 10), (1, 2, 1)]
-            sage: sorted(g.min_spanning_tree(algorithm='Boruvka', weight_function=weight))
+            sage: Graph(g.min_spanning_tree(algorithm='NetworkX', weight_function=weight)).edges(sort=True)   # needs networkx
             [(0, 2, 10), (1, 2, 1)]
 
         If the graph is directed, it is transformed into an undirected graph::
 
             sage: g = digraphs.Circuit(3)
-            sage: sorted(g.min_spanning_tree(weight_function=weight))
+            sage: Graph(g.min_spanning_tree(weight_function=weight)).edges(sort=True)
             [(0, 2, None), (1, 2, None)]
-            sage: sorted(g.to_undirected().min_spanning_tree(weight_function=weight))
+            sage: Graph(g.to_undirected().min_spanning_tree(weight_function=weight)).edges(sort=True)
             [(0, 2, None), (1, 2, None)]
 
         If at least an edge weight is not convertible to a float, an error is
@@ -4841,6 +4856,17 @@ def min_spanning_tree(self,
 
             sage: graphs.EmptyGraph().min_spanning_tree()
             []
+
+        Check that the method is robust to incomparable vertices::
+
+            sage: G = Graph([(1, 2, 10), (1, 'a', 1), ('a', 'b', 1), ('b', 2, 1)])
+            sage: E = G.min_spanning_tree(algorithm='Prim_Boost', by_weight=True)
+            sage: E = G.min_spanning_tree(algorithm='Prim_fringe', by_weight=True)
+            sage: E = G.min_spanning_tree(algorithm='Prim_edge', by_weight=True)
+            sage: E = G.min_spanning_tree(algorithm='Kruskal_Boost', by_weight=True)
+            sage: E = G.min_spanning_tree(algorithm='Filter_Kruskal', by_weight=True)
+            sage: E = G.min_spanning_tree(algorithm='Boruvka', by_weight=True)
+            sage: E = G.min_spanning_tree(algorithm='NetworkX', by_weight=True)         # needs networkx
         """
         if not self.order():
             return []
@@ -5136,9 +5162,9 @@ def cycle_basis(self, output='vertex'):
             sage: [sorted(c) for c in G.cycle_basis()]                                  # needs networkx
             [['Hey', 'Really ?', 'Wuuhuu'], [0, 2], [0, 1, 2]]
             sage: [sorted(c) for c in G.cycle_basis(output='edge')]                     # needs networkx
-            [[('Hey', 'Wuuhuu', None),
-              ('Really ?', 'Hey', None),
-              ('Wuuhuu', 'Really ?', None)],
+            [[('Hey', 'Really ?', None),
+              ('Really ?', 'Wuuhuu', None),
+              ('Wuuhuu', 'Hey', None)],
              [(0, 2, 'a'), (2, 0, 'b')],
              [(0, 2, 'b'), (1, 0, 'c'), (2, 1, 'd')]]
 

src/sage/graphs/spanning_tree.pyx

@@ -241,6 +241,13 @@ def kruskal(G, by_weight=True, weight_function=None, check_weight=False, check=F
         Traceback (most recent call last):
         ...
         ValueError: the input graph must be undirected
+
+    Check that the method is robust to incomparable vertices::
+
+        sage: G = Graph([(1, 2, 10), (1, 'a', 1), ('a', 'b', 1), ('b', 2, 1)])
+        sage: E = kruskal(G, by_weight=True)
+        sage: sum(w for _, _, w in E)
+        3
     """
     return list(kruskal_iterator(G, by_weight=by_weight, weight_function=weight_function,
                                  check_weight=check_weight, check=check))
@@ -313,6 +320,13 @@ def kruskal_iterator(G, by_weight=True, weight_function=None, check_weight=False
         Traceback (most recent call last):
         ...
         ValueError: the input graph must be undirected
+
+    Check that the method is robust to incomparable vertices::
+
+        sage: G = Graph([(1, 2, 10), (1, 'a', 1), ('a', 'b', 1), ('b', 2, 1)])
+        sage: E = list(kruskal_iterator(G, by_weight=True))
+        sage: sum(w for _, _, w in E)
+        3
     """
     from sage.graphs.graph import Graph
     if not isinstance(G, Graph):
@@ -382,6 +396,14 @@ def kruskal_iterator_from_edges(edges, union_find, by_weight=True,
         sage: union_set = DisjointSet(G)
         sage: next(kruskal_iterator_from_edges(G.edges(sort=False), union_set, by_weight=G.weighted()))
         (1, 6, 10)
+
+    Check that the method is robust to incomparable vertices::
+
+        sage: G = Graph([(1, 2, 10), (1, 'a', 1), ('a', 'b', 1), ('b', 2, 1)])
+        sage: union_set = DisjointSet(G)
+        sage: E = list(kruskal_iterator_from_edges(G.edges(sort=False), union_set, by_weight=True))
+        sage: sum(w for _, _, w in E)
+        3
     """
     # We sort edges, as specified.
     if weight_function is not None:
@@ -472,6 +494,15 @@ def filter_kruskal(G, threshold=10000, by_weight=True, weight_function=None,
 
         sage: filter_kruskal(Graph(2), check=True)
         []
+
+    TESTS:
+
+    Check that the method is robust to incomparable vertices::
+
+        sage: G = Graph([(1, 2, 10), (1, 'a', 1), ('a', 'b', 1), ('b', 2, 1)])
+        sage: E = filter_kruskal(G, by_weight=True)
+        sage: sum(w for _, _, w in E)
+        3
     """
     return list(filter_kruskal_iterator(G, threshold=threshold,
                                         by_weight=by_weight, weight_function=weight_function,
@@ -563,6 +594,13 @@ def filter_kruskal_iterator(G, threshold=10000, by_weight=True, weight_function=
 
         sage: len(list(filter_kruskal_iterator(graphs.HouseGraph(), threshold=1)))
         4
+
+    Check that the method is robust to incomparable vertices::
+
+        sage: G = Graph([(1, 2, 10), (1, 'a', 1), ('a', 'b', 1), ('b', 2, 1)])
+        sage: E = list(filter_kruskal_iterator(G, by_weight=True))
+        sage: sum(w for _, _, w in E)
+        3
     """
     from sage.graphs.graph import Graph
     if not isinstance(G, Graph):
@@ -776,6 +814,13 @@ def boruvka(G, by_weight=True, weight_function=None, check_weight=True, check=Fa
         Traceback (most recent call last):
         ...
         ValueError: the input graph must be undirected
+
+    Check that the method is robust to incomparable vertices::
+
+        sage: G = Graph([(1, 2, 10), (1, 'a', 1), ('a', 'b', 1), ('b', 2, 1)])
+        sage: E = boruvka(G, by_weight=True)
+        sage: sum(w for _, _, w in E)
+        3
     """
     from sage.graphs.graph import Graph
     if not isinstance(G, Graph):
@@ -985,6 +1030,13 @@ def random_spanning_tree(G, output_as_graph=False, by_weight=False, weight_funct
         Traceback (most recent call last):
         ...
         ValueError: works only for non-empty connected graphs
+
+    Check that the method is robust to incomparable vertices::
+
+        sage: G = Graph([(1, 2, 10), (1, 'a', 1), ('a', 'b', 1), ('b', 2, 1)])
+        sage: T = G.random_spanning_tree(by_weight=True, output_as_graph=True)
+        sage: T.is_tree()
+        True
     """
     from sage.misc.prandom import randint
     from sage.misc.prandom import random
@@ -1113,6 +1165,12 @@ def spanning_trees(g, labels=False):
         Traceback (most recent call last):
         ...
         ValueError: this method is for undirected graphs only
+
+    Check that the method is robust to incomparable vertices::
+
+        sage: G = Graph([(1, 2, 10), (1, 'a', 1), ('a', 'b', 1), ('b', 2, 1)])
+        sage: len(list(G.spanning_trees(labels=False)))
+        4
     """
     from sage.graphs.graph import Graph
     if not isinstance(g, Graph):
@@ -1257,6 +1315,14 @@ def edge_disjoint_spanning_trees(G, k, by_weight=False, weight_function=None, ch
         Traceback (most recent call last):
         ...
         ValueError: this method is for undirected graphs only
+
+    Check that the method is robust to incomparable vertices::
+
+        sage: G = Graph()
+        sage: G.add_clique([0, 1, 2, 'a', 'b'])
+        sage: F = G.edge_disjoint_spanning_trees(k=2)
+        sage: len(F)
+        2
     """
     if G.is_directed():
         raise ValueError("this method is for undirected graphs only")

src/sage/matroids/utilities.py

@@ -572,26 +572,28 @@ def lift_cross_ratios(A, lift_map=None):
 
     G = Graph([((r, 0), (c, 1), (r, c)) for r, c in A.nonzero_positions()])
     # write the entries of (a scaled version of) A as products of cross ratios of A
-    T = set()
+    T = Graph()
     for C in G.connected_components_subgraphs():
-        T.update(C.min_spanning_tree())
+        T.add_edges(C.min_spanning_tree())
     # - fix a tree of the support graph G to units (= empty dict, product of 0 terms)
-    F = {entry[2]: dict() for entry in T}
-    W = set(G.edge_iterator()) - set(T)
-    H = G.subgraph(edges=T)
+    F = {entry: dict() for entry in T.edge_labels()}
+    W = set(G.edge_iterator()) - set(T.edge_iterator())
+    H = G.subgraph(edges=T.edge_iterator())
     while W:
         # - find an edge in W to process, closing a circuit in H which is induced in G
         edge = W.pop()
         path = H.shortest_path(edge[0], edge[1])
+        path_s = set(path)
         retry = True
         while retry:
             retry = False
             for edge2 in W:
-                if edge2[0] in path and edge2[1] in path:
+                if edge2[0] in path_s and edge2[1] in path_s:
                     W.add(edge)
                     edge = edge2
                     W.remove(edge)
                     path = H.shortest_path(edge[0], edge[1])
+                    path_s = set(path)
                     retry = True
                     break
         entry = edge[2]