diff --git a/BGL/include/CGAL/boost/graph/helpers.h b/BGL/include/CGAL/boost/graph/helpers.h
index ee0247330bd..6b87731ad9b 100644
--- a/BGL/include/CGAL/boost/graph/helpers.h
+++ b/BGL/include/CGAL/boost/graph/helpers.h
@@ -945,6 +945,9 @@ void swap_edges(const typename boost::graph_traits<FaceGraph>::halfedge_descript
  * \link MutableHalfedgeGraph `remove_edge()`\endlink, and
  * \link MutableFaceGraph `remove_face()`\endlink, for each vertex, edge, and face.
  *
+ * Note that some graphs have a specialized version of this function to improve
+ * complexity.
+ *
  * @warning This function does not perform anything more than what is advertised above. It is
  * up to the user to e.g. clean garbage or remove internal property maps (if relevant, and desired).
  *
diff --git a/Polyhedron/include/CGAL/boost/graph/graph_traits_Polyhedron_3.h b/Polyhedron/include/CGAL/boost/graph/graph_traits_Polyhedron_3.h
index 8cff560f0a6..33a45710ab1 100644
--- a/Polyhedron/include/CGAL/boost/graph/graph_traits_Polyhedron_3.h
+++ b/Polyhedron/include/CGAL/boost/graph/graph_traits_Polyhedron_3.h
@@ -310,6 +310,12 @@ set_halfedge(typename boost::graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::ver
   static_cast<Sneak&>(*v).set_halfedge(h);
 }
 
+template<class Gt, class I, CGAL_HDS_PARAM_, class A>
+void
+remove_all_elements(CGAL::Polyhedron_3<Gt,I,HDS,A>& g)
+{
+  g.clear();
+}
 
 //
 // HalfedgeGraph
diff --git a/Surface_mesh/include/CGAL/boost/graph/graph_traits_Surface_mesh.h b/Surface_mesh/include/CGAL/boost/graph/graph_traits_Surface_mesh.h
index d044c38a557..1a05f71bb9c 100644
--- a/Surface_mesh/include/CGAL/boost/graph/graph_traits_Surface_mesh.h
+++ b/Surface_mesh/include/CGAL/boost/graph/graph_traits_Surface_mesh.h
@@ -472,7 +472,6 @@ remove_vertex(typename boost::graph_traits<CGAL::Surface_mesh<P> >::vertex_descr
   sm.remove_vertex(v);
 }
 
-
 template <typename P>
 void
 remove_edge(typename boost::graph_traits<CGAL::Surface_mesh<P> >::vertex_descriptor u,
@@ -491,7 +490,6 @@ remove_edge(typename boost::graph_traits<CGAL::Surface_mesh<P> >::edge_descripto
   sm.remove_edge(e);
 }
 
-
 template <typename P>
 void
 remove_edge(typename boost::graph_traits<CGAL::Surface_mesh<P> >::edge_iterator eiter,
@@ -508,6 +506,13 @@ remove_face(typename boost::graph_traits<CGAL::Surface_mesh<P> >::face_descripto
   sm.remove_face(f);
 }
 
+template<typename P>
+void
+remove_all_elements(CGAL::Surface_mesh<P>& sm)
+{
+  sm.clear_without_removing_property_maps();
+}
+
 template<typename P>
 typename boost::graph_traits<CGAL::Surface_mesh<P> >::face_descriptor
 add_face(CGAL::Surface_mesh<P>& sm)