diff --git a/Surface_mesh_topology/examples/Surface_mesh_topology/facewidth_on_unweighted_map.cpp b/Surface_mesh_topology/examples/Surface_mesh_topology/facewidth_on_unweighted_map.cpp
index c939af3aac9..8d761ae0bd4 100644
--- a/Surface_mesh_topology/examples/Surface_mesh_topology/facewidth_on_unweighted_map.cpp
+++ b/Surface_mesh_topology/examples/Surface_mesh_topology/facewidth_on_unweighted_map.cpp
@@ -7,10 +7,10 @@
 #include <CGAL/Path_on_surface.h>
 #include <CGAL/draw_linear_cell_complex.h>
 
-using LCC_3          =CGAL::Linear_cell_complex_for_combinatorial_map<2, 3>;
-using CST            =CGAL::Surface_mesh_topology::Curves_on_surface_topology<LCC_3>;
-using Path_on_surface=CGAL::Surface_mesh_topology::Path_on_surface<LCC_3>;
-using Dart_handle    =LCC_3::Dart_handle;
+using LCC_3            =CGAL::Linear_cell_complex_for_combinatorial_map<2, 3>;
+using CST              =CGAL::Surface_mesh_topology::Curves_on_surface_topology<LCC_3>;
+using Path_on_surface  =CGAL::Surface_mesh_topology::Path_on_surface<LCC_3>;
+using Dart_const_handle=LCC_3::Dart_const_handle;
 
 struct Draw_functor : public CGAL::DefaultDrawingFunctorLCC
 {
@@ -71,7 +71,7 @@ int main(int argc, char* argv[])
   std::cout<<"File '"<<filename<<"' loaded. Finding the facewidth..."<<std::endl;
   
   CST cst(lcc, true);
-  std::vector<Dart_handle> cycle = cst.compute_facewidth(true);
+  std::vector<Dart_const_handle> cycle=cst.compute_facewidth(true);
 
   if (cycle.size()==0)
   { std::cout<<"  Cannot find such cycle. Stop."<<std::endl; }
diff --git a/Surface_mesh_topology/include/CGAL/Curves_on_surface_topology.h b/Surface_mesh_topology/include/CGAL/Curves_on_surface_topology.h
index 1a5caf4d1e0..98dbf357666 100644
--- a/Surface_mesh_topology/include/CGAL/Curves_on_surface_topology.h
+++ b/Surface_mesh_topology/include/CGAL/Curves_on_surface_topology.h
@@ -34,7 +34,8 @@ public:
   // Types for shortest noncontractible cycle
   using Shortest_noncontractible_cycle=typename internal::Shortest_noncontractible_cycle<Mesh>;
   using Facewidth                     =typename internal::Facewidth<Mesh>;
-  using Dart_handle                   =typename Shortest_noncontractible_cycle::Original_dart_const_handle;
+  using Dart_const_handle             =typename Shortest_noncontractible_cycle::Original_dart_const_handle;
+  using halfedge_descriptor           =Dart_const_handle ; // To be compatible with BGL
 
   // Constructor
   Curves_on_surface_topology(const Mesh& amesh, bool /* display_time */=false) :
@@ -114,28 +115,19 @@ public:
 
   template <class WeightFunctor>
   Path_on_surface<Mesh> compute_shortest_noncontractible_cycle_with_basepoint
-  (Dart_handle dh, const WeightFunctor& wf, bool display_time=false) const
+  (Dart_const_handle dh, const WeightFunctor& wf, bool display_time=false) const
   {
     compute_shortest_noncontractible_cycle_representation(display_time);
     return m_shortest_noncontractible_cycle->compute_cycle(dh, NULL, wf, display_time);
   }
 
   Path_on_surface<Mesh> compute_shortest_noncontractible_cycle_with_basepoint
-  (Dart_handle dh, bool display_time=false) const
+  (Dart_const_handle dh, bool display_time=false) const
   {
     compute_shortest_noncontractible_cycle_representation(display_time);
     return m_shortest_noncontractible_cycle->compute_cycle(dh, display_time);
   }
 
-  bool is_facewidth_representation_computed() const
-  { return m_facewidth!=nullptr; }
-  
-  void compute_facewidth_representation(bool display_time=false) const
-  {
-    if (m_facewidth==nullptr)
-    { m_facewidth=std::make_unique<Facewidth>(m_original_mesh, display_time); }
-  }
-
   template <class WeightFunctor>
   Path_on_surface<Mesh> compute_edgewidth(const WeightFunctor& wf, bool display_time=false) const
   {
@@ -149,7 +141,16 @@ public:
     return m_shortest_noncontractible_cycle->compute_edgewidth(display_time);
   }
 
-  std::vector<Dart_handle> compute_facewidth(bool display_time=false) const
+  bool is_facewidth_representation_computed() const
+  { return m_facewidth!=nullptr; }
+
+  void compute_facewidth_representation(bool display_time=false) const
+  {
+    if (m_facewidth==nullptr)
+    { m_facewidth=std::make_unique<Facewidth>(m_original_mesh, display_time); }
+  }
+
+  std::vector<Dart_const_handle> compute_facewidth(bool display_time=false) const
   {
     compute_facewidth_representation(display_time);
     return m_facewidth->compute_facewidth(display_time);
diff --git a/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Facewidth.h b/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Facewidth.h
index de302b84530..f4a44b81fcf 100644
--- a/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Facewidth.h
+++ b/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Facewidth.h
@@ -31,7 +31,7 @@ public:
   using Mesh=Mesh_;
 
   using Original_map_wrapper=internal::Generic_map_selector<Mesh>;
-  using Original_dart_const_handle=typename Original_map_wrapper::Dart_handle_original; // TODO SOLVE const problem with copy
+  using Original_dart_const_handle=typename Original_map_wrapper::Dart_const_handle_original;
 
   using Local_map        =typename Original_map_wrapper::Generic_map;
   using Dart_handle      =typename Local_map::Dart_handle;
diff --git a/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Generic_map_selector.h b/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Generic_map_selector.h
index f411e94cd79..d4502177556 100644
--- a/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Generic_map_selector.h
+++ b/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Generic_map_selector.h
@@ -38,15 +38,15 @@ namespace internal {
   struct SNC_for_generalized_map {
     using Mesh_original              = Mesh_;
     using Generic_map                = CGAL::Generalized_map<2, Shortest_noncontractible_cycle_local_map_items>;
-    using Dart_handle_original       = typename Mesh_original::Dart_handle;
     using Dart_const_handle_original = typename Mesh_original::Dart_const_handle;
     using Copy_to_origin_map         = boost::unordered_map<typename Generic_map::Dart_handle, 
-                                                            Dart_handle_original>;
-    using Origin_to_copy_map         = boost::unordered_map<Dart_handle_original, 
+                                                            Dart_const_handle_original>;
+    using Origin_to_copy_map         = boost::unordered_map<Dart_const_handle_original,
                                                             typename Generic_map::Dart_handle>;
 
-    static void copy(Generic_map& target, Mesh_original& source,
-                     Origin_to_copy_map& origin_to_copy, Copy_to_origin_map& copy_to_origin,
+    static void copy(Generic_map& target, const Mesh_original& source,
+                     Origin_to_copy_map& origin_to_copy,
+                     Copy_to_origin_map& copy_to_origin,
                      Generic_map::size_type mark_perforated)
     {
       target.copy(source, &origin_to_copy, &copy_to_origin, true, mark_perforated);
@@ -57,14 +57,13 @@ namespace internal {
   struct SNC_for_combinatorial_map {
     using Mesh_original              = Mesh_;
     using Generic_map                = CGAL::Combinatorial_map<2, Shortest_noncontractible_cycle_local_map_items>;
-    using Dart_handle_original       = typename Mesh_original::Dart_handle;
     using Dart_const_handle_original = typename Mesh_original::Dart_const_handle;
     using Copy_to_origin_map         = boost::unordered_map<typename Generic_map::Dart_handle, 
-                                                            Dart_handle_original>;
-    using Origin_to_copy_map         = boost::unordered_map<Dart_handle_original, 
+                                                            Dart_const_handle_original>;
+    using Origin_to_copy_map         = boost::unordered_map<Dart_const_handle_original,
                                                             typename Generic_map::Dart_handle>;
 
-    static void copy(Generic_map& target, Mesh_original& source,
+    static void copy(Generic_map& target, const Mesh_original& source,
                      Origin_to_copy_map& origin_to_copy, Copy_to_origin_map& copy_to_origin,
                      Generic_map::size_type mark_perforated)
     {
@@ -76,14 +75,13 @@ namespace internal {
   struct Generic_map_selector {
     using Mesh_original              = Mesh_;
     using Generic_map                = CGAL::Combinatorial_map<2, Shortest_noncontractible_cycle_local_map_items>;
-    using Dart_handle_original       = typename boost::graph_traits<Mesh_original>::halfedge_descriptor;
     using Dart_const_handle_original = typename boost::graph_traits<Mesh_original>::halfedge_descriptor;
     using Copy_to_origin_map         = boost::unordered_map<typename Generic_map::Dart_handle, 
-                                                            Dart_handle_original>;
-    using Origin_to_copy_map         = boost::unordered_map<Dart_handle_original, 
+                                                            Dart_const_handle_original>;
+    using Origin_to_copy_map         = boost::unordered_map<Dart_const_handle_original,
                                                             typename Generic_map::Dart_handle>;
 
-    static void copy(Generic_map& target, Mesh_original& source,
+    static void copy(Generic_map& target, const Mesh_original& source,
                      Origin_to_copy_map& origin_to_copy, Copy_to_origin_map& copy_to_origin,
                      Generic_map::size_type mark_perforated)
     {
diff --git a/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Shortest_noncontractible_cycle.h b/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Shortest_noncontractible_cycle.h
index 4fbfadef888..84827fc5c8c 100644
--- a/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Shortest_noncontractible_cycle.h
+++ b/Surface_mesh_topology/include/CGAL/Surface_mesh_topology/internal/Shortest_noncontractible_cycle.h
@@ -43,7 +43,7 @@ public:
   using Mesh=Mesh_;
 
   using Original_map_wrapper=internal::Generic_map_selector<Mesh>;
-  using Original_dart_const_handle=typename Original_map_wrapper::Dart_handle_original; // TODO SOLVE const problem with copy
+  using Original_dart_const_handle=typename Original_map_wrapper::Dart_const_handle_original;
 
   using Local_map        =typename Original_map_wrapper::Generic_map;
   using Dart_handle      =typename Local_map::Dart_handle;