gh-39287: add parameter `immutable` to transitive closure methods in `sage/graphs/generic_graph.py`
    
Following #39280 and discussions in #39177, we add parameter `immutable`
to methods related to transitive closure in
`sage/graphs/generic_graph.py` and `sage/graphs/generic_graph_pyx.pyx`:
- `transitive_closure`. We also fix the use of parameter `loops` that
was previously ignored
- `transitive_reduction`
- `transitive_reduction_acyclic`
- `is_transitively_reduced`. Here we ensure that the method accepts
immutable digraphs.


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### ⌛ Dependencies

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URL: #39287
Reported by: David Coudert
Reviewer(s): Frédéric Chapoton

Author: Release Manager

src/sage/categories/kahler_algebras.py

@@ -142,33 +142,33 @@ def hodge_riemann_relations(self, k):
                 sage: ch = matroids.Uniform(4, 6).chow_ring(QQ, False)
                 sage: ch.hodge_riemann_relations(1)
                 Quadratic form in 36 variables over Rational Field with coefficients:
-                [ 3 -1 -1 3 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 3 ]
-                [ * 3 -1 3 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 3 ]
-                [ * * 3 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 3 ]
+                [ 3 -1 -1 3 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 3 ]
+                [ * 3 -1 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * 3 3 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 3 ]
                 [ * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
-                [ * * * * 3 -1 3 -1 3 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 3 ]
-                [ * * * * * 3 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 3 ]
+                [ * * * * 3 -1 3 -1 -1 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * * * * 3 3 -1 3 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 3 ]
                 [ * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
-                [ * * * * * * * 3 3 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * * * * * * 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 3 ]
                 [ * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
                 [ * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
-                [ * * * * * * * * * * 3 -1 3 -1 -1 3 -1 -1 3 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 ]
-                [ * * * * * * * * * * * 3 3 -1 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * * * * * * * * * 3 -1 3 -1 -1 3 -1 -1 -1 3 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * * * * * * * * * * 3 3 -1 3 -1 -1 -1 3 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 ]
                 [ * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
-                [ * * * * * * * * * * * * * 3 3 3 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 -1 -1 3 ]
+                [ * * * * * * * * * * * * * 3 3 3 -1 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 -1 -1 3 ]
                 [ * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
                 [ * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
-                [ * * * * * * * * * * * * * * * * 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 -1 -1 -1 3 ]
+                [ * * * * * * * * * * * * * * * * 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 3 ]
                 [ * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
                 [ * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
                 [ * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
-                [ * * * * * * * * * * * * * * * * * * * * 3 -1 3 -1 3 -1 -1 -1 -1 3 -1 -1 -1 3 -1 3 ]
-                [ * * * * * * * * * * * * * * * * * * * * * 3 3 -1 -1 3 -1 -1 3 -1 -1 -1 3 -1 -1 3 ]
+                [ * * * * * * * * * * * * * * * * * * * * 3 -1 3 -1 3 -1 -1 -1 -1 3 -1 -1 -1 -1 3 3 ]
+                [ * * * * * * * * * * * * * * * * * * * * * 3 3 -1 -1 3 -1 -1 3 -1 -1 -1 -1 3 -1 3 ]
                 [ * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
-                [ * * * * * * * * * * * * * * * * * * * * * * * 3 3 3 -1 3 -1 -1 -1 3 -1 -1 -1 3 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * 3 3 3 -1 3 -1 -1 -1 -1 3 -1 -1 3 ]
                 [ * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
                 [ * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ]
-                [ * * * * * * * * * * * * * * * * * * * * * * * * * * 3 3 3 3 -1 -1 -1 -1 3 3 ]
+                [ * * * * * * * * * * * * * * * * * * * * * * * * * * 3 3 3 3 -1 3 -1 -1 -1 3 ]
                 [ * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 -1 ]
                 [ * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 -1 ]
                 [ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 -1 -1 -1 -1 -1 -1 ]

src/sage/combinat/posets/lattices.py

@@ -4877,9 +4877,9 @@ def quotient(self, congruence, labels='tuple'):
         parts_H = [sorted([self._element_to_vertex(e) for e in part]) for
                    part in congruence]
         minimal_vertices = [part[0] for part in parts_H]
-        H = self._hasse_diagram.transitive_closure().subgraph(minimal_vertices).transitive_reduction()
+        H = self._hasse_diagram.transitive_closure().subgraph(minimal_vertices).transitive_reduction(immutable=False)
         if labels == 'integer':
-            H.relabel(list(range(len(minimal_vertices))))
+            H.relabel()
             return LatticePoset(H)
         part_dict = {m[0]: [self._vertex_to_element(x) for x in m] for m
                      in parts_H}

src/sage/graphs/generic_graph.py

@@ -20516,7 +20516,7 @@ def disjunctive_product(self, other, immutable=None):
         return GT([vertices, edges], format='vertices_and_edges',
                   loops=loops, immutable=immutable)
 
-    def transitive_closure(self, loops=True):
+    def transitive_closure(self, loops=None, immutable=None):
         r"""
         Return the transitive closure of the (di)graph.
 
@@ -20528,11 +20528,18 @@ def transitive_closure(self, loops=True):
         acyclic graph is a directed acyclic graph representing the full partial
         order.
 
-        .. NOTE::
+        INPUT:
 
-            If the (di)graph allows loops, its transitive closure will by
-            default have one loop edge per vertex. This can be prevented by
-            disallowing loops in the (di)graph (``self.allow_loops(False)``).
+        - ``loops`` -- boolean (default: ``None``); whether to allow loops in
+          the returned (di)graph. By default (``None``), if the (di)graph allows
+          loops, its transitive closure will have one loop edge per vertex. This
+          can be prevented by disallowing loops in the (di)graph
+          (``self.allow_loops(False)``).
+
+        - ``immutable`` -- boolean (default: ``None``); whether to create a
+          mutable/immutable transitive closure. ``immutable=None`` (default)
+          means that the (di)graph and its transitive closure will behave the
+          same way.
 
         EXAMPLES::
 
@@ -20560,21 +20567,49 @@ def transitive_closure(self, loops=True):
             sage: G.transitive_closure().loop_edges(labels=False)
             [(0, 0), (1, 1), (2, 2)]
 
-        ::
+        Check the behavior of parameter ``loops``::
 
             sage: G = graphs.CycleGraph(3)
             sage: G.transitive_closure().loop_edges(labels=False)
             []
+            sage: G.transitive_closure(loops=True).loop_edges(labels=False)
+            [(0, 0), (1, 1), (2, 2)]
             sage: G.allow_loops(True)
             sage: G.transitive_closure().loop_edges(labels=False)
             [(0, 0), (1, 1), (2, 2)]
+            sage: G.transitive_closure(loops=False).loop_edges(labels=False)
+            []
+
+        Check the behavior of parameter ``immutable``::
+
+            sage: G = Graph([(0, 1)])
+            sage: G.transitive_closure().is_immutable()
+            False
+            sage: G.transitive_closure(immutable=True).is_immutable()
+            True
+            sage: G = Graph([(0, 1)], immutable=True)
+            sage: G.transitive_closure().is_immutable()
+            True
+            sage: G.transitive_closure(immutable=False).is_immutable()
+            False
         """
-        G = copy(self)
-        G.name('Transitive closure of ' + self.name())
-        G.add_edges(((u, v) for u in G for v in G.breadth_first_search(u)), loops=None)
-        return G
+        name = f"Transitive closure of {self.name()}"
+        if immutable is None:
+            immutable = self.is_immutable()
+        if loops is None:
+            loops = self.allows_loops()
+        if loops:
+            edges = ((u, v) for u in self for v in self.depth_first_search(u))
+        else:
+            edges = ((u, v) for u in self for v in self.depth_first_search(u) if u != v)
+        if self.is_directed():
+            from sage.graphs.digraph import DiGraph as GT
+        else:
+            from sage.graphs.graph import Graph as GT
+        return GT([self, edges], format='vertices_and_edges', loops=loops,
+                  immutable=immutable, name=name)
 
-    def transitive_reduction(self):
+    def transitive_reduction(self, immutable=None):
         r"""
         Return a transitive reduction of a graph.
 
@@ -20587,6 +20622,13 @@ def transitive_reduction(self):
         A transitive reduction of a complete graph is a tree. A transitive
         reduction of a tree is itself.
 
+        INPUT:
+
+        - ``immutable`` -- boolean (default: ``None``); whether to create a
+          mutable/immutable transitive closure. ``immutable=None`` (default)
+          means that the (di)graph and its transitive closure will behave the
+          same way.
+
         EXAMPLES::
 
             sage: g = graphs.PathGraph(4)
@@ -20595,38 +20637,72 @@ def transitive_reduction(self):
             sage: g = graphs.CompleteGraph(5)
             sage: h = g.transitive_reduction(); h.size()
             4
+            sage: (2*g).transitive_reduction().size()
+            8
             sage: g = DiGraph({0: [1, 2], 1: [2, 3, 4, 5], 2: [4, 5]})
             sage: g.transitive_reduction().size()
             5
+            sage: (2*g).transitive_reduction().size()
+            10
+
+        TESTS:
+
+        Check the behavior of parameter ``immutable``::
+
+            sage: G = Graph([(0, 1)])
+            sage: G.transitive_reduction().is_immutable()
+            False
+            sage: G.transitive_reduction(immutable=True).is_immutable()
+            True
+            sage: G = Graph([(0, 1)], immutable=True)
+            sage: G.transitive_reduction().is_immutable()
+            True
+            sage: G = DiGraph([(0, 1), (1, 2), (2, 0)])
+            sage: G.transitive_reduction().is_immutable()
+            False
+            sage: G.transitive_reduction(immutable=True).is_immutable()
+            True
+            sage: G = DiGraph([(0, 1), (1, 2), (2, 0)], immutable=True)
+            sage: G.transitive_reduction().is_immutable()
+            True
         """
+        if immutable is None:
+            immutable = self.is_immutable()
+
         if self.is_directed():
             if self.is_directed_acyclic():
                 from sage.graphs.generic_graph_pyx import transitive_reduction_acyclic
-                return transitive_reduction_acyclic(self)
+                return transitive_reduction_acyclic(self, immutable=immutable)
 
-            G = copy(self)
+            G = self.copy(immutable=False)
             G.allow_multiple_edges(False)
             n = G.order()
-            for e in G.edges(sort=False):
+            for e in list(G.edges(sort=False)):
                 # Try deleting the edge, see if we still have a path between
                 # the vertices.
                 G.delete_edge(e)
                 if G.distance(e[0], e[1]) > n:
                     # oops, we shouldn't have deleted it
                     G.add_edge(e)
+            if immutable:
+                return G.copy(immutable=True)
             return G
 
         # The transitive reduction of each connected component of an
         # undirected graph is a spanning tree
-        from sage.graphs.graph import Graph
         if self.is_connected():
-            return Graph(self.min_spanning_tree(weight_function=lambda e: 1))
-        G = Graph(list(self))
-        for cc in self.connected_components(sort=False):
-            if len(cc) > 1:
-                edges = self.subgraph(cc).min_spanning_tree(weight_function=lambda e: 1)
-                G.add_edges(edges)
-        return G
+            CC = [self]
+        else:
+            CC = (self.subgraph(c)
+                  for c in self.connected_components(sort=False) if len(c) > 1)
+
+        def edges():
+            for g in CC:
+                yield from g.min_spanning_tree(weight_function=lambda e: 1)
+
+        from sage.graphs.graph import Graph
+        return Graph([self, edges()], format='vertices_and_edges',
+                     immutable=immutable)
 
     def is_transitively_reduced(self):
         r"""
@@ -20648,13 +20724,27 @@ def is_transitively_reduced(self):
             sage: d = DiGraph({0: [1, 2], 1: [2], 2: []})
             sage: d.is_transitively_reduced()
             False
+
+        TESTS:
+
+        Check the behavior of the method for immutable (di)graphs::
+
+            sage: G = DiGraph([(0, 1), (1, 2), (2, 0)], immutable=True)
+            sage: G.is_transitively_reduced()
+            True
+            sage: G = DiGraph(graphs.CompleteGraph(4), immutable=True)
+            sage: G.is_transitively_reduced()
+            False
+            sage: G = Graph([(0, 1), (2, 3)], immutable=True)
+            sage: G.is_transitively_reduced()
+            True
         """
         if self.is_directed():
             if self.is_directed_acyclic():
                 return self == self.transitive_reduction()
 
             from sage.rings.infinity import Infinity
-            G = copy(self)
+            G = self.copy(immutable=False)
             for e in self.edge_iterator():
                 G.delete_edge(e)
                 if G.distance(e[0], e[1]) == Infinity:

src/sage/graphs/generic_graph_pyx.pyx

@@ -1585,20 +1585,37 @@ cpdef tuple find_hamiltonian(G, long max_iter=100000, long reset_bound=30000,
     return (True, output)
 
 
-def transitive_reduction_acyclic(G):
+def transitive_reduction_acyclic(G, immutable=None):
     r"""
     Return the transitive reduction of an acyclic digraph.
 
     INPUT:
 
     - ``G`` -- an acyclic digraph
 
+    - ``immutable`` -- boolean (default: ``None``); whether to create a
+      mutable/immutable transitive closure. ``immutable=None`` (default) means
+      that the (di)graph and its transitive closure will behave the same way.
+
     EXAMPLES::
 
         sage: from sage.graphs.generic_graph_pyx import transitive_reduction_acyclic
         sage: G = posets.BooleanLattice(4).hasse_diagram()
         sage: G == transitive_reduction_acyclic(G.transitive_closure())
         True
+
+    TESTS:
+
+    Check the behavior of parameter ``immutable``::
+
+        sage: G = DiGraph([(0, 1)])
+        sage: transitive_reduction_acyclic(G).is_immutable()
+        False
+        sage: transitive_reduction_acyclic(G, immutable=True).is_immutable()
+        True
+        sage: G = DiGraph([(0, 1)], immutable=True)
+        sage: transitive_reduction_acyclic(G).is_immutable()
+        True
     """
     cdef int  n = G.order()
     cdef dict v_to_int = {vv: i for i, vv in enumerate(G)}
@@ -1642,10 +1659,13 @@ def transitive_reduction_acyclic(G):
             if binary_matrix_get(closure, u, v):
                 useful_edges.append((uu, vv))
 
+    if immutable is None:
+        immutable = G.is_immutable()
+
     from sage.graphs.digraph import DiGraph
-    reduced = DiGraph()
-    reduced.add_edges(useful_edges)
-    reduced.add_vertices(linear_extension)
+    reduced = DiGraph([linear_extension, useful_edges],
+                      format='vertices_and_edges',
+                      immutable=immutable)
 
     binary_matrix_free(closure)