Trac #27557: spanning_trees does not enumerate spanning trees with ed…

…ge labels

Currently spanning_trees() method does not enumerate spanning trees with
their labels , it makes their labels None. This tickets aims to return
the correct labels of the edges.

URL: https://trac.sagemath.org/27557
Reported by: gh-rajat1433
Ticket author(s): Rajat Mittal
Reviewer(s): David Coudert

src/sage/graphs/graph.py

@@ -1431,14 +1431,19 @@ def is_directed(self):
 
 
     @doc_index("Connectivity, orientations, trees")
-    def spanning_trees(self):
+    def spanning_trees(self, labels=False):
         """
         Returns a list of all spanning trees.
 
         If the graph is disconnected, returns the empty list.
 
         Uses the Read-Tarjan backtracking algorithm [RT75]_.
 
+        INPUT:
+
+        - ``labels`` -- boolean (default: ``False``); whether to return edges
+          labels in the spanning trees or not
+
         EXAMPLES::
 
             sage: G = Graph([(1,2),(1,2),(1,3),(1,3),(2,3),(1,4)], multiedges=True)
@@ -1473,14 +1478,25 @@ def spanning_trees(self):
              Graph on 6 vertices,
              Graph on 6 vertices]
 
+        Edges of the spanning trees can be labeled or unlabeled (:trac:27557)::
+
+            sage: g = Graph([(1,2,2),(1,2,1),(1,2,4),(1,4,5)],multiedges=True)
+            sage: l = g.spanning_trees(labels=True)
+            sage: l[0].edges()
+            [(1, 2, 4), (1, 4, 5)]
+            sage: l[1].edges()
+            [(1, 2, 1), (1, 4, 5)]
+            sage: l[2].edges()
+            [(1, 2, 2), (1, 4, 5)]
+
         REFERENCES:
 
         .. [RT75] Read, R. C. and Tarjan, R. E.
           Bounds on Backtrack Algorithms for Listing Cycles, Paths, and Spanning Trees
           Networks, Volume 5 (1975), numer 3, pages 237-252.
         """
 
-        def _recursive_spanning_trees(G, forest):
+        def _recursive_spanning_trees(G, forest, labels):
             """
             Returns all the spanning trees of G containing forest
             """
@@ -1491,13 +1507,13 @@ def _recursive_spanning_trees(G, forest):
                 return [forest.copy()]
             else:
                 # Pick an edge e from G-forest
-                for e in G.edge_iterator(labels=False):
+                for e in G.edge_iterator(labels=labels):
                     if not forest.has_edge(e):
                         break
 
                 # 1) Recursive call with e removed from G
                 G.delete_edge(e)
-                trees = _recursive_spanning_trees(G, forest)
+                trees = _recursive_spanning_trees(G, forest, labels)
                 G.add_edge(e)
 
                 # 2) Recursive call with e include in forest
@@ -1514,7 +1530,7 @@ def _recursive_spanning_trees(G, forest):
                 # Actual call
                 forest.add_edge(e)
                 G.delete_edges(B)
-                trees.extend(_recursive_spanning_trees(G, forest))
+                trees.extend(_recursive_spanning_trees(G, forest, labels))
                 G.add_edges(B)
                 forest.delete_edge(e)
 
@@ -1524,7 +1540,7 @@ def _recursive_spanning_trees(G, forest):
             forest = Graph()
             forest.add_vertices(self.vertex_iterator())
             forest.add_edges(self.bridges())
-            return _recursive_spanning_trees(Graph(self, immutable=False, loops=False), forest)
+            return _recursive_spanning_trees(Graph(self, immutable=False, loops=False), forest, labels)
         else:
             return []