Trac #24113: Add test for cactus graph

This patch adds a function to check if a graph is cactus graph.

URL: https://trac.sagemath.org/24113
Reported by: jmantysalo
Ticket author(s): Jori Mäntysalo
Reviewer(s): David Coudert

src/sage/graphs/graph.py

@@ -1798,6 +1798,59 @@ def is_forest(self, certificate=False, output='vertex'):
                 if not isit:
                     return (False, cycle)
 
+    @doc_index("Graph properties")
+    def is_cactus(self):
+        """
+        Check whether the graph is cactus graph.
+
+        A graph is called *cactus graph* if it is connected and every pair of
+        simple cycles have at most one common vertex.
+
+        There are other definitions, see :wikipedia:`Cactus_graph`.
+
+        EXAMPLES::
+
+            sage: g = Graph({1: [2], 2: [3, 4], 3: [4, 5, 6, 7], 8: [3, 5], 9: [6, 7]})
+            sage: g.is_cactus()
+            True
+
+            sage: c6 = graphs.CycleGraph(6)
+            sage: naphthalene = c6 + c6
+            sage: naphthalene.is_cactus()  # Not connected
+            False
+            sage: naphthalene.merge_vertices([0, 6])
+            sage: naphthalene.is_cactus()
+            True
+            sage: naphthalene.merge_vertices([1, 7])
+            sage: naphthalene.is_cactus()
+            False
+
+        TESTS::
+
+            sage: all(graphs.PathGraph(i).is_cactus() for i in range(5))
+            True
+
+            sage: Graph('Fli@?').is_cactus()
+            False
+        """
+        self._scream_if_not_simple()
+
+        # Special cases
+        if self.order() < 4:
+            return True
+
+        # Every cactus graph is outerplanar, and outerplanar
+        # graphs have limited number of edges.
+        if self.size() > self.order()*2-3:
+            return False
+
+        if not self.is_connected():
+            return False
+
+        # the number of faces is 1 plus the number of blocks of order > 2
+        B = self.blocks_and_cut_vertices()[0]
+        return len(self.faces()) == sum(1 for b in B if len(b) > 2) + 1
+
     @doc_index("Graph properties")
     def is_biconnected(self):
         """

src/sage/graphs/isgci.py

@@ -1061,6 +1061,7 @@ def _XML_to_dict(root):
 graph_classes.BinaryTrees = GraphClass("BinaryTrees", "gc_847")
 graph_classes.Bipartite = GraphClass("Bipartite", "gc_69", recognition_function = lambda x:x.is_bipartite())
 graph_classes.Block = GraphClass("Block", "gc_93", recognition_function = lambda x:x.is_block_graph())
+graph_classes.Cactus = GraphClass("Cactus", "gc_108", recognition_function = lambda x:x.is_cactus())
 graph_classes.Chordal = GraphClass("Chordal", "gc_32", recognition_function = lambda x:x.is_chordal())
 graph_classes.ClawFree = GraphClass("Claw-free", "gc_62")
 graph_classes.CoGraph = GraphClass("CoGraph", "gc_151", recognition_function = lambda x:x.is_cograph())