diff --git a/Combinatorial_map/include/CGAL/Cell_attribute.h b/Combinatorial_map/include/CGAL/Cell_attribute.h
index 796b71ba629..df3a4b0d686 100644
--- a/Combinatorial_map/include/CGAL/Cell_attribute.h
+++ b/Combinatorial_map/include/CGAL/Cell_attribute.h
@@ -32,25 +32,25 @@ template<unsigned int, class, class>
 class Combinatorial_map_storage_1;
 
 template<unsigned int, class, class>
-class Combinatorial_map_storage_2;
+class Combinatorial_map_storage_with_index;
 
 template<unsigned int, class, class>
 class Generalized_map_storage_1;
 
 template<unsigned int, class, class>
-class Generalized_map_storage_2;
+class Generalized_map_storage_with_index;
 
 template<unsigned int, unsigned int, class, class, class>
 class CMap_linear_cell_complex_storage_1;
 
 template<unsigned int, unsigned int, class, class, class>
-class CMap_linear_cell_complex_storage_2;
+class CMap_linear_cell_complex_storage_with_index;
 
 template<unsigned int, unsigned int, class, class, class>
 class GMap_linear_cell_complex_storage_1;
 
 template<unsigned int, unsigned int, class, class, class>
-class GMap_linear_cell_complex_storage_2;
+class GMap_linear_cell_complex_storage_with_index;
 
 namespace internal {
 
@@ -111,25 +111,25 @@ struct Init_id;
     friend class Combinatorial_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Combinatorial_map_storage_2;
+    friend class Combinatorial_map_storage_with_index;
 
     template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Generalized_map_storage_2;
+    friend class Generalized_map_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class CMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class CMap_linear_cell_complex_storage_2;
+    friend class CMap_linear_cell_complex_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class GMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class GMap_linear_cell_complex_storage_2;
+    friend class GMap_linear_cell_complex_storage_with_index;
 
     template<class, class>
     friend struct internal::Init_id;
@@ -236,25 +236,25 @@ struct Init_id;
     friend class Combinatorial_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Combinatorial_map_storage_2;
+    friend class Combinatorial_map_storage_with_index;
 
     template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Generalized_map_storage_2;
+    friend class Generalized_map_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class CMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class CMap_linear_cell_complex_storage_2;
+    friend class CMap_linear_cell_complex_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class GMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class GMap_linear_cell_complex_storage_2;
+    friend class GMap_linear_cell_complex_storage_with_index;
 
     template<class, class>
     friend struct internal::Init_id;
@@ -368,25 +368,25 @@ struct Init_id;
     friend class Combinatorial_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Combinatorial_map_storage_2;
+    friend class Combinatorial_map_storage_with_index;
 
     template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Generalized_map_storage_2;
+    friend class Generalized_map_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class CMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class CMap_linear_cell_complex_storage_2;
+    friend class CMap_linear_cell_complex_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class GMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class GMap_linear_cell_complex_storage_2;
+    friend class GMap_linear_cell_complex_storage_with_index;
 
   public:
     typedef Tag_                             Supports_cell_dart;
@@ -423,25 +423,25 @@ struct Init_id;
     friend class Combinatorial_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Combinatorial_map_storage_2;
+    friend class Combinatorial_map_storage_with_index;
 
     template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Generalized_map_storage_2;
+    friend class Generalized_map_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class CMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class CMap_linear_cell_complex_storage_2;
+    friend class CMap_linear_cell_complex_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class GMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class GMap_linear_cell_complex_storage_2;
+    friend class GMap_linear_cell_complex_storage_with_index;
 
   public:
     typedef Cell_attribute<Refs, Info_, Tag_, OnMerge, OnSplit, WithID> Self;
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_fwd.h b/Combinatorial_map/include/CGAL/Combinatorial_map_fwd.h
index 65f02ff17fd..cd9432defe8 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_fwd.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_fwd.h
@@ -24,7 +24,7 @@ template<unsigned int d_, class Items_, class Alloc_>
 class Combinatorial_map_storage_1;
 
 template<unsigned int d_, class Items_, class Alloc_>
-class Combinatorial_map_storage_2;
+class Combinatorial_map_storage_with_index;
 
 namespace internal
 {
@@ -38,7 +38,7 @@ template<>
 struct Default_storage_for_cmap_when_tag<CGAL::Tag_true>
 {
   template<unsigned int d_, class Items_, class Alloc_>
-  using type=Combinatorial_map_storage_2<d_, Items_, Alloc_>;
+  using type=Combinatorial_map_storage_with_index<d_, Items_, Alloc_>;
 };
 
 BOOST_MPL_HAS_XXX_TRAIT_NAMED_DEF(Has_use_index_tag_cmap,Use_index,false)
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_storages_with_index.h b/Combinatorial_map/include/CGAL/Combinatorial_map_storages_with_index.h
index be89f501f2e..abd5f6200e0 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_storages_with_index.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_storages_with_index.h
@@ -34,10 +34,10 @@ namespace CGAL {
 
   // Storage with combinatorial maps using index
   template<unsigned int d_, class Items_, class Alloc_>
-  class Combinatorial_map_storage_2
+  class Combinatorial_map_storage_with_index
   {
   public:
-    using Self=Combinatorial_map_storage_2<d_, Items_, Alloc_>;
+    using Self=Combinatorial_map_storage_with_index<d_, Items_, Alloc_>;
     using Use_index=CGAL::Tag_true;
     using Concurrent_tag=typename internal::Get_concurrent_tag<Items_>::type;
 
@@ -65,7 +65,7 @@ namespace CGAL {
     typedef typename Dart_container::size_type size_type;
 
     typedef Dart_index Null_descriptor_type;
-    static Null_descriptor_type null_descriptor;
+    static const Index_type null_descriptor=Dart_container::null_descriptor;
 
     using Type_for_compact_container=Index_type;
 
@@ -108,7 +108,7 @@ namespace CGAL {
     template<int i>
     using Attribute_const_handle=Attribute_const_descriptor<i>;
 
-    static Null_descriptor_type null_handle;
+    static const Index_type null_handle=null_descriptor;
 
     /// Number of marks
     static const size_type NB_MARKS = 32;
@@ -154,7 +154,7 @@ namespace CGAL {
     Dart_const_range& darts() const { return mdarts_range; }
     //**************************************************************************
 
-    Combinatorial_map_storage_2() : mdarts_range(*this)
+    Combinatorial_map_storage_with_index() : mdarts_range(*this)
     {}
 
     void init_storage()
@@ -164,7 +164,7 @@ namespace CGAL {
       Dart_index local_null_dart_descriptor = mdarts.emplace();
       if(local_null_dart_descriptor!=0)
       {
-        std::cerr<<"[ERROR] fatal in Combinatorial_map_storage_2::init_storage"
+        std::cerr<<"[ERROR] fatal in Combinatorial_map_storage_with_index::init_storage"
                  <<std::endl;
         CGAL_assertion(false);
       }
@@ -470,8 +470,10 @@ namespace CGAL {
 
   public:
     /// Void dart. A dart d is i-free if beta_i(d)=null_dart_descriptor.
-    static Dart_index null_dart_descriptor; //=0;
-    static Dart_descriptor null_dart_handle; // Deprecated: kept for backward compatibility
+    static const Index_type null_dart_descriptor=0;
+
+    // Deprecated: kept for backward compatibility
+    static const Index_type null_dart_handle=null_dart_descriptor;
 
   protected:
     /// Dart container.
@@ -485,26 +487,6 @@ namespace CGAL {
     typename Helper::Attribute_containers mattribute_containers;
   };
 
-  /// null_dart_descriptor
-  template<unsigned int d_, class Items_, class Alloc_>
-  typename Combinatorial_map_storage_2<d_, Items_, Alloc_>::Dart_index
-  Combinatorial_map_storage_2<d_, Items_, Alloc_>::null_dart_descriptor(0);
-
-  /// null_descriptor
-  template<unsigned int d_, class Items_, class Alloc_>
-  typename Combinatorial_map_storage_2<d_, Items_, Alloc_>::Null_descriptor_type
-      Combinatorial_map_storage_2<d_, Items_, Alloc_>::
-  null_descriptor((std::numeric_limits<Index_type>::max)());
-
-  template<unsigned int d_, class Items_, class Alloc_>
-  typename Combinatorial_map_storage_2<d_, Items_, Alloc_>::Dart_index
-  Combinatorial_map_storage_2<d_, Items_, Alloc_>::null_dart_handle(0);
-
-  template<unsigned int d_, class Items_, class Alloc_>
-  typename Combinatorial_map_storage_2<d_, Items_, Alloc_>::Null_descriptor_type
-      Combinatorial_map_storage_2<d_, Items_, Alloc_>::
-  null_handle((std::numeric_limits<Index_type>::max)());
-
 } // namespace CGAL
 
 #if defined(BOOST_GCC)
diff --git a/Combinatorial_map/include/CGAL/Compact_container_with_index.h b/Combinatorial_map/include/CGAL/Compact_container_with_index.h
index 1bcf372713d..efbc2349a80 100644
--- a/Combinatorial_map/include/CGAL/Compact_container_with_index.h
+++ b/Combinatorial_map/include/CGAL/Compact_container_with_index.h
@@ -97,6 +97,8 @@ class Free_list_management<CC_with_index, CGAL::Tag_true, CGAL::Tag_true>
   using size_type=typename CC_with_index::size_type;
 
 public:
+  static const size_type null_descriptor=std::numeric_limits<size_type>::max();
+
   Free_list_management(CC_with_index& cc_with_index):
     m_cc_with_index(cc_with_index)
   {}
@@ -191,6 +193,8 @@ class Free_list_management<CC_with_index, CGAL::Tag_false, CGAL::Tag_true>
   using Traits=Compact_container_with_index_traits <T, size_type>;
 
 public:
+  static const size_type null_descriptor=std::numeric_limits<size_type>::max();
+
   Free_list_management(CC_with_index& cc_with_index):
     m_cc_with_index(cc_with_index)
   {}
@@ -200,7 +204,7 @@ public:
     m_used.assign(m_cc_with_index.capacity(), false);
     m_free_list=0;
     for(size_type i=0; i<static_cast<size_type>(m_cc_with_index.capacity()); ++i)
-    { Traits::size_t(m_cc_with_index[i])=i+1; }
+    { Traits::set_size_t(m_cc_with_index[i], i+1); }
     // Next of the last element is capacity() which is the "nullptr".
   }
 
@@ -210,7 +214,7 @@ public:
     CGAL_assertion(m_free_list==old_size); // Previous container was full
     m_used.resize(m_cc_with_index.capacity(), false);
     for(size_type i=0; i<static_cast<size_type>(m_cc_with_index.capacity()); ++i)
-    { Traits::size_t(m_cc_with_index[i])=i+1; }
+    { Traits::set_size_t(m_cc_with_index[i], i+1); }
     // Nothing to do with m_free_list, because it was equal to old_size.
   }
 
@@ -234,7 +238,7 @@ public:
   {
     CGAL_assertion(i<m_cc_with_index.capacity());
     m_used[i]=false;
-    Traits::size_t(m_cc_with_index[i])=m_free_list;
+    Traits::set_size_t(m_cc_with_index[i], m_free_list);
     m_free_list=i;
   }
 
@@ -249,11 +253,12 @@ public:
     CGAL_assertion(!is_empty());
     size_type res=m_free_list;
     m_free_list=Traits::size_t(m_cc_with_index[res]);
+    m_used[res]=true;
     return res;
   }
 
   void copy_special_data(const T& src, T& dest)
-  { Traits::size_t(dest)=Traits::size_t(src); }
+  { Traits::set_size_t(dest, Traits::size_t(src)); }
 
 protected:
   CC_with_index&    m_cc_with_index;
@@ -271,6 +276,8 @@ class Free_list_management<CC_with_index, CGAL::Tag_false, CGAL::Tag_false>
   using Traits=Compact_container_with_index_traits <T, size_type>;
 
 public:
+  static const size_type null_descriptor=std::numeric_limits<size_type>::max()/2;
+
   Free_list_management(CC_with_index& cc_with_index):
     m_cc_with_index(cc_with_index)
   {}
@@ -327,7 +334,7 @@ public:
   }
 
   void copy_special_data(const T& src, T& dest)
-  { Traits::size_t(dest)=Traits::size_t(src); }
+  { Traits::set_size_t(dest, Traits::size_t(src)); }
 
 protected:
   // Definition of the bit squatting :
@@ -351,6 +358,11 @@ protected:
   static size_type static_get_val(const T& e)
   { return (Traits::size_t(e) & ~mask_type); }
 
+  // set the value of the element and its type
+  static void static_set_type(T& e, Type t)
+  { Traits::set_size_t(e, static_get_val(e) |
+                       ( ((size_type)t) <<(nbbits_size_type_m1))); }
+
   // set the value of the element and its type
   static void static_set_val(T& e, size_type v, Type t)
   { Traits::set_size_t(e, v | ( ((size_type)t) <<(nbbits_size_type_m1))); }
@@ -368,7 +380,7 @@ public:
   using Self=Index_for_cc_with_index<Index_type>;
   using size_type=Index_type;
 
-  /// Constructor. Default construction creates a kind of "NULL" index.
+  /// Constructor.
   Index_for_cc_with_index(size_type idx=(std::numeric_limits<size_type>::max)())
     : m_idx(idx)
   {}
@@ -382,9 +394,11 @@ public:
   Index_for_cc_with_index(const Index2& idx): m_idx(static_cast<size_t>(idx))
   {}
 
-  /// return whether the handle is valid
-  bool is_valid() const
-  { return m_idx != (std::numeric_limits<size_type>::max)(); }
+  bool operator==(const Self& n) const
+  { return m_idx==n.m_idx; }
+
+  bool operator==(size_type n) const
+  { return m_idx==n; }
 
   /// Increment the internal index. This operations does not
   /// guarantee that the index is valid or undeleted after the
@@ -447,6 +461,11 @@ public:
   typedef std::reverse_iterator<const_iterator>     const_reverse_iterator;
 
   using Index=Index_for_cc_with_index<IndexType>;
+  using TFree_list_management=Free_list_management
+                             <Self, CGAL::Tag_false, CGAL::Tag_true>;
+
+  static const size_type null_descriptor=TFree_list_management::null_descriptor;
+
   friend class internal::CC_iterator_with_index<Self, false>;
   friend class internal::CC_iterator_with_index<Self, true>;
 
@@ -728,12 +747,12 @@ private:
     free_list.init();
   }
 
-  allocator_type   alloc;
-  size_type        capacity_;
-  size_type        size_;
-  size_type        block_size;
-  pointer          all_items;
-  Free_list_management<Self, CGAL::Tag_true, CGAL::Tag_true> free_list;
+  allocator_type        alloc;
+  size_type             capacity_;
+  size_type             size_;
+  size_type             block_size;
+  pointer               all_items;
+  TFree_list_management free_list;
 };
 
 /*template < class T, class Allocator, class Increment_policy, class IndexType >
diff --git a/Combinatorial_map/include/CGAL/Dart.h b/Combinatorial_map/include/CGAL/Dart.h
index 9246004d586..3ae2a9d449e 100644
--- a/Combinatorial_map/include/CGAL/Dart.h
+++ b/Combinatorial_map/include/CGAL/Dart.h
@@ -33,25 +33,25 @@ namespace CGAL {
   class Combinatorial_map_storage_1;
 
   template<unsigned int, class, class>
-  class Combinatorial_map_storage_2;
+  class Combinatorial_map_storage_with_index;
 
   template<unsigned int, class, class>
   class Generalized_map_storage_1;
 
   template<unsigned int, class, class>
-  class Generalized_map_storage_2;
+  class Generalized_map_storage_with_index;
 
   template<unsigned int, unsigned int, class, class, class>
   class CMap_linear_cell_complex_storage_1;
 
   template<unsigned int, unsigned int, class, class, class>
-  class CMap_linear_cell_complex_storage_2;
+  class CMap_linear_cell_complex_storage_with_index;
 
   template<unsigned int, unsigned int, class, class, class>
   class GMap_linear_cell_complex_storage_1;
 
   template<unsigned int, unsigned int, class, class, class>
-  class GMap_linear_cell_complex_storage_2;
+  class GMap_linear_cell_complex_storage_with_index;
 
   namespace internal {
 
@@ -88,35 +88,35 @@ namespace CGAL {
     friend class Combinatorial_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Combinatorial_map_storage_2;
+    friend class Combinatorial_map_storage_with_index;
 
     template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Generalized_map_storage_2;
+    friend class Generalized_map_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class CMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class CMap_linear_cell_complex_storage_2;
+    friend class CMap_linear_cell_complex_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class GMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class GMap_linear_cell_complex_storage_2;
+    friend class GMap_linear_cell_complex_storage_with_index;
 
     template<class, class>
     friend struct internal::Init_id;
 
-    typedef Dart_without_info<d,Refs, WithId> Self;
-    typedef typename Refs::Dart_descriptor        Dart_descriptor;
-    typedef typename Refs::size_type          size_type;
-    typedef typename Refs::Dart_const_descriptor  Dart_const_descriptor;
-    typedef typename Refs::Helper             Helper;
-    typedef WithId                            Has_id;
+    typedef Dart_without_info<d,Refs, WithId>    Self;
+    typedef typename Refs::Dart_descriptor       Dart_descriptor;
+    typedef typename Refs::size_type             size_type;
+    typedef typename Refs::Dart_const_descriptor Dart_const_descriptor;
+    typedef typename Refs::Helper                Helper;
+    typedef WithId                               Has_id;
     using Type_for_compact_container=typename Refs::Type_for_compact_container;
 
     /// Typedef for attributes
@@ -251,25 +251,25 @@ namespace CGAL {
     friend class Combinatorial_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Combinatorial_map_storage_2;
+    friend class Combinatorial_map_storage_with_index;
 
     template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Generalized_map_storage_2;
+    friend class Generalized_map_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class CMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class CMap_linear_cell_complex_storage_2;
+    friend class CMap_linear_cell_complex_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class GMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class GMap_linear_cell_complex_storage_2;
+    friend class GMap_linear_cell_complex_storage_with_index;
 
     typedef Dart<d, Refs, Info_, WithID> Self;
     typedef Info_                        Info;
diff --git a/Generalized_map/include/CGAL/Generalized_map_fwd.h b/Generalized_map/include/CGAL/Generalized_map_fwd.h
index 6da380457b9..4febe72d5c2 100644
--- a/Generalized_map/include/CGAL/Generalized_map_fwd.h
+++ b/Generalized_map/include/CGAL/Generalized_map_fwd.h
@@ -22,7 +22,7 @@ template<unsigned int d_, class Items_, class Alloc_>
 class Generalized_map_storage_1;
 
 template<unsigned int d_, class Items_, class Alloc_>
-class Generalized_map_storage_2;
+class Generalized_map_storage_with_index;
 
 struct Generic_map_min_items;
 
@@ -38,7 +38,7 @@ template<>
 struct Default_storage_for_gmap_when_tag<CGAL::Tag_true>
 {
   template<unsigned int d_, class Items_, class Alloc_>
-  using type=Generalized_map_storage_2<d_, Items_, Alloc_>;
+  using type=Generalized_map_storage_with_index<d_, Items_, Alloc_>;
 };
 
 BOOST_MPL_HAS_XXX_TRAIT_NAMED_DEF(Has_use_index_tag_gmap,Use_index,false)
diff --git a/Generalized_map/include/CGAL/Generalized_map_storages_with_index.h b/Generalized_map/include/CGAL/Generalized_map_storages_with_index.h
index c0f0624760d..d19bced77cc 100644
--- a/Generalized_map/include/CGAL/Generalized_map_storages_with_index.h
+++ b/Generalized_map/include/CGAL/Generalized_map_storages_with_index.h
@@ -34,10 +34,10 @@ namespace CGAL {
 
   // Storage of darts with compact container, alpha with indices
   template<unsigned int d_, class Items_, class Alloc_>
-  class Generalized_map_storage_2
+  class Generalized_map_storage_with_index
   {
   public:
-    using Self=Generalized_map_storage_2<d_, Items_, Alloc_>;
+    using Self=Generalized_map_storage_with_index<d_, Items_, Alloc_>;
     using Use_index=CGAL::Tag_true;
     using Concurrent_tag=typename internal::Get_concurrent_tag<Items_>::type;
 
@@ -439,13 +439,13 @@ namespace CGAL {
 
   /// null_descriptor
   template<unsigned int d_, class Items_, class Alloc_>
-  typename Generalized_map_storage_2<d_, Items_, Alloc_>::Null_descriptor_type
-      Generalized_map_storage_2<d_, Items_, Alloc_>::
+  typename Generalized_map_storage_with_index<d_, Items_, Alloc_>::Null_descriptor_type
+      Generalized_map_storage_with_index<d_, Items_, Alloc_>::
   null_descriptor((std::numeric_limits<Index_type>::max)());
 
   template<unsigned int d_, class Items_, class Alloc_>
-  typename Generalized_map_storage_2<d_, Items_, Alloc_>::Null_descriptor_type
-      Generalized_map_storage_2<d_, Items_, Alloc_>::
+  typename Generalized_map_storage_with_index<d_, Items_, Alloc_>::Null_descriptor_type
+      Generalized_map_storage_with_index<d_, Items_, Alloc_>::
   null_handle((std::numeric_limits<Index_type>::max)());
 
 } // namespace CGAL
diff --git a/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages_with_index.h b/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages_with_index.h
index 0a121c9ff72..a2a94774979 100644
--- a/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages_with_index.h
+++ b/Linear_cell_complex/include/CGAL/CMap_linear_cell_complex_storages_with_index.h
@@ -33,16 +33,16 @@ namespace CGAL {
   }
 
   // Storage of darts with compact container, alpha using index
-  // Copy of Combinatorial_map_storage_2 and add new types related
+  // Copy of Combinatorial_map_storage_with_index and add new types related
   // to geometry (not possible to inherith because we use Self type
   // as template parameter of Dart_wrapper. If we inherit, Self is not
   // the correct type).
   template<unsigned int d_, unsigned int ambient_dim, class Traits_,
            class Items_, class Alloc_>
-  class CMap_linear_cell_complex_storage_2
+  class CMap_linear_cell_complex_storage_with_index
   {
   public:
-    using Self=CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
+    using Self=CMap_linear_cell_complex_storage_with_index<d_, ambient_dim, Traits_,
                                           Items_, Alloc_>;
     using Use_index=CGAL::Tag_true;
     using Concurrent_tag=typename internal::Get_concurrent_tag<Items_>::type;
@@ -75,7 +75,7 @@ namespace CGAL {
     typedef typename Dart_container::size_type size_type;
 
     typedef Dart_index Null_descriptor_type;
-    static Null_descriptor_type null_descriptor;
+    static const Index_type null_descriptor=Dart_container::null_descriptor;
 
     using Type_for_compact_container=Index_type;
 
@@ -129,7 +129,7 @@ namespace CGAL {
     using Vertex_attribute_handle=Vertex_attribute_descriptor;
     using Vertex_attribute_const_handle=Vertex_attribute_const_descriptor;
 
-    static Null_descriptor_type null_handle;
+    static const Index_type null_handle=null_descriptor;
 
     /// Number of marks
     static const size_type NB_MARKS = 32;
@@ -175,7 +175,7 @@ namespace CGAL {
     Dart_const_range& darts() const { return mdarts_range; }
     //**************************************************************************
 
-    CMap_linear_cell_complex_storage_2() : mdarts_range(*this)
+    CMap_linear_cell_complex_storage_with_index() : mdarts_range(*this)
     {}
 
     void init_storage()
@@ -185,7 +185,7 @@ namespace CGAL {
       Dart_index local_null_dart_descriptor = mdarts.emplace();
       if(local_null_dart_descriptor!=0)
       {
-        std::cerr<<"[ERROR] fatal in CMap_linear_cell_complex_storage_2::init_storage"
+        std::cerr<<"[ERROR] fatal in CMap_linear_cell_complex_storage_with_index::init_storage"
                  <<std::endl;
         CGAL_assertion(false);
       }
@@ -505,8 +505,10 @@ namespace CGAL {
 
   public:
     /// Void dart. A dart d is i-free if beta_i(d)=null_dart_descriptor.
-    static Dart_index null_dart_descriptor; //=0;
-    static Dart_descriptor null_dart_handle; // Deprecated: kept for backward compatibility
+    static const Index_type null_dart_descriptor=0;
+
+    // Deprecated: kept for backward compatibility
+    static const Index_type null_dart_handle=null_dart_descriptor;
 
   protected:
     /// Dart container.
@@ -520,34 +522,6 @@ namespace CGAL {
     typename Helper::Attribute_containers mattribute_containers;
   };
 
-  /// null_dart_descriptor
-  template<unsigned int d_,  unsigned int ambient_dim, class Traits_,
-           class Items_, class Alloc_>
-  typename CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
-  Dart_index CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
-  null_dart_descriptor(0);
-
-  /// null_descriptor
-  template<unsigned int d_, unsigned int ambient_dim, class Traits_,
-           class Items_, class Alloc_>
-  typename CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
-  Null_descriptor_type CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
-                                                      Items_, Alloc_>::
-  null_descriptor((std::numeric_limits<Index_type>::max)());
-
-  template<unsigned int d_,  unsigned int ambient_dim, class Traits_,
-           class Items_, class Alloc_>
-  typename CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
-  Dart_index CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
-  null_dart_handle(0);
-
-  template<unsigned int d_, unsigned int ambient_dim, class Traits_,
-           class Items_, class Alloc_>
-  typename CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
-  Null_descriptor_type CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
-                                                      Items_, Alloc_>::
-  null_handle((std::numeric_limits<Index_type>::max)());
-
 } // namespace CGAL
 
 #if defined(BOOST_GCC)
diff --git a/Linear_cell_complex/include/CGAL/Cell_attribute_with_point.h b/Linear_cell_complex/include/CGAL/Cell_attribute_with_point.h
index a791f7f6088..17c17d1b87b 100644
--- a/Linear_cell_complex/include/CGAL/Cell_attribute_with_point.h
+++ b/Linear_cell_complex/include/CGAL/Cell_attribute_with_point.h
@@ -69,25 +69,25 @@ namespace CGAL {
     friend class Combinatorial_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Combinatorial_map_storage_2;
+    friend class Combinatorial_map_storage_with_index;
 
     template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Generalized_map_storage_2;
+    friend class Generalized_map_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class CMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class CMap_linear_cell_complex_storage_2;
+    friend class CMap_linear_cell_complex_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class GMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class GMap_linear_cell_complex_storage_2;
+    friend class GMap_linear_cell_complex_storage_with_index;
 
   public:
     typedef Cell_attribute_with_point<LCC, Info_, Tag, Functor_on_merge_,
@@ -157,25 +157,25 @@ namespace CGAL {
     friend class Combinatorial_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Combinatorial_map_storage_2;
+    friend class Combinatorial_map_storage_with_index;
 
     template<unsigned int, class, class>
     friend class Generalized_map_storage_1;
 
     template<unsigned int, class, class>
-    friend class Generalized_map_storage_2;
+    friend class Generalized_map_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class CMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class CMap_linear_cell_complex_storage_2;
+    friend class CMap_linear_cell_complex_storage_with_index;
 
     template<unsigned int, unsigned int, class, class, class>
     friend class GMap_linear_cell_complex_storage_1;
 
     template<unsigned int, unsigned int, class, class, class>
-    friend class GMap_linear_cell_complex_storage_2;
+    friend class GMap_linear_cell_complex_storage_with_index;
 
   public:
     typedef Cell_attribute<LCC, void, Tag,
diff --git a/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages_with_index.h b/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages_with_index.h
index 59d37d2b544..6b81113139f 100644
--- a/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages_with_index.h
+++ b/Linear_cell_complex/include/CGAL/GMap_linear_cell_complex_storages_with_index.h
@@ -33,16 +33,16 @@ namespace CGAL {
   }
 
   // Storage of darts with compact container, alpha using index
-  // Copy of Generalized_map_storage_2 and add new types related
+  // Copy of Generalized_map_storage_with_index and add new types related
   // to geometry (not possible to inherith because we use Self type
   // as template parameter of Dart_wrapper. If we inherit, Self is not
   // the correct type).
   template<unsigned int d_, unsigned int ambient_dim, class Traits_,
            class Items_, class Alloc_>
-  class GMap_linear_cell_complex_storage_2
+  class GMap_linear_cell_complex_storage_with_index
   {
   public:
-    using Self=GMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
+    using Self=GMap_linear_cell_complex_storage_with_index<d_, ambient_dim, Traits_,
                                           Items_, Alloc_>;
     using Use_index=CGAL::Tag_true;
     using Concurrent_tag=typename internal::Get_concurrent_tag<Items_>::type;
@@ -462,8 +462,8 @@ namespace CGAL {
   /// null_descriptor
   template<unsigned int d_, unsigned int ambient_dim, class Traits_,
            class Items_, class Alloc_>
-  typename GMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_, Items_, Alloc_>::
-  Null_descriptor_type GMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
+  typename GMap_linear_cell_complex_storage_with_index<d_, ambient_dim, Traits_, Items_, Alloc_>::
+  Null_descriptor_type GMap_linear_cell_complex_storage_with_index<d_, ambient_dim, Traits_,
                                              Items_, Alloc_>::
   null_descriptor((std::numeric_limits<Index_type>::max)());
 
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex_fwd.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex_fwd.h
index 013fffeefa1..2a807a71b9c 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex_fwd.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex_fwd.h
@@ -27,11 +27,11 @@ class GMap_linear_cell_complex_storage_1;
 
 template<unsigned int d_, unsigned int ambient_dim,
          class Traits_, class Items_, class Alloc_>
-class CMap_linear_cell_complex_storage_2;
+class CMap_linear_cell_complex_storage_with_index;
 
 template<unsigned int d_, unsigned int ambient_dim,
          class Traits_, class Items_, class Alloc_>
-class GMap_linear_cell_complex_storage_2;
+class GMap_linear_cell_complex_storage_with_index;
 
 template <unsigned int d>
 struct LCC_default_kernel;
@@ -57,7 +57,7 @@ struct Default_storage_for_lcc_cmap_when_tag<CGAL::Tag_true>
 {
   template<unsigned int d_, unsigned int ambient_dim,
            class Traits_, class Items_, class Alloc_>
-  using type=CMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
+  using type=CMap_linear_cell_complex_storage_with_index<d_, ambient_dim, Traits_,
   Items_, Alloc_>;
 };
 
@@ -92,7 +92,7 @@ struct Default_storage_for_lcc_gmap_when_tag<CGAL::Tag_true>
 {
   template<unsigned int d_, unsigned int ambient_dim,
            class Traits_, class Items_, class Alloc_>
-  using type=GMap_linear_cell_complex_storage_2<d_, ambient_dim, Traits_,
+  using type=GMap_linear_cell_complex_storage_with_index<d_, ambient_dim, Traits_,
   Items_, Alloc_>;
 };