Trac #24118: Add quick check for planarity

Functions to check if a graph is planar or circular planar should first
check if the graph has "too many" edges compared to number of vertices
and then return `False` directly.

URL: https://trac.sagemath.org/24118
Reported by: jmantysalo
Ticket author(s): Jori Mäntysalo
Reviewer(s): David Coudert

src/sage/graphs/generic_graph.py

@@ -4239,7 +4239,20 @@ def is_planar(self, on_embedding=None, kuratowski=False, set_embedding=False, se
 
             sage: posets.BooleanLattice(3).cover_relations_graph().is_planar()
             True
+
+        Corner cases::
+
+            sage: graphs.EmptyGraph().is_planar()
+            True
+            sage: Graph(1).is_planar()
+            True
         """
+        # Quick check first
+        if (on_embedding is None and not kuratowski and not set_embedding and not set_pos
+            and not self.allows_loops() and not self.allows_multiple_edges()):
+            if self.order() > 4 and self.size() > 3*self.order()-6:
+                return False
+
         if self.has_multiple_edges() or self.has_loops():
             if set_embedding or (on_embedding is not None) or set_pos:
                 raise NotImplementedError("Cannot compute with embeddings of multiple-edged or looped graphs.")
@@ -4267,7 +4280,7 @@ def is_planar(self, on_embedding=None, kuratowski=False, set_embedding=False, se
             return planar
 
     def is_circular_planar(self, on_embedding=None, kuratowski=False,
-                           set_embedding=True, boundary = None,
+                           set_embedding=True, boundary=None,
                            ordered=False, set_pos=False):
         """
         Tests whether the graph is circular planar (outerplanar)
@@ -4373,10 +4386,26 @@ def is_circular_planar(self, on_embedding=None, kuratowski=False,
 
             sage: K23.is_circular_planar(set_embedding=True, boundary = [0,2,1,3])
             True
+
+        TESTS:
+
+        Corner cases::
+
+            sage: graphs.EmptyGraph().is_circular_planar()
+            True
+            sage: Graph(1).is_circular_planar()
+            True
         """
         if ordered and boundary is None:
             raise ValueError("boundary must be set when ordered is True.")
 
+        # Quick check first
+        if (on_embedding is None and not kuratowski and set_embedding and
+            boundary is None and not ordered and not set_pos and
+            not self.allows_loops() and not self.allows_multiple_edges()):
+            if self.order() > 3 and self.size() > 2*self.order()-3:
+                return False
+
         if boundary is None:
             boundary = self