diff --git a/Combinatorial_map/include/CGAL/Cell_const_iterators.h b/Combinatorial_map/include/CGAL/Cell_const_iterators.h
index 7cf76d7f4ed..3d60c035c41 100644
--- a/Combinatorial_map/include/CGAL/Cell_const_iterators.h
+++ b/Combinatorial_map/include/CGAL/Cell_const_iterators.h
@@ -91,12 +91,13 @@ namespace CGAL {
     /// Constructor with a dart in parameter (for end iterator).
     CMap_one_dart_per_cell_const_iterator(const Map_& amap, 
                                           Dart_const_handle adart): 
-      Base(amap)
-    { this->set_current_dart(adart); }
+      Base(amap, adart)
+    {}
     /// Constructor from non const version.
     CMap_one_dart_per_cell_const_iterator
     (const CMap_one_dart_per_cell_iterator<Map_,i,dim,false>& it):
-      Base(*const_cast<const Map_*>(it.get_combinatorial_map()))
+      Base(*const_cast<const Map_*>(it.get_combinatorial_map()),
+           it.get_first_dart())
     {}
   };
   //****************************************************************************
diff --git a/Combinatorial_map/include/CGAL/Cell_iterators.h b/Combinatorial_map/include/CGAL/Cell_iterators.h
index fea03a3418d..b29c741955d 100644
--- a/Combinatorial_map/include/CGAL/Cell_iterators.h
+++ b/Combinatorial_map/include/CGAL/Cell_iterators.h
@@ -48,7 +48,7 @@ namespace CGAL {
   /* Class CMap_cell_iterator<Map,Ite,i,dim,Tag_true>: to iterate onto the
    * cells incident to the given iterator which uses mark.
    */
-  template <typename Map_,typename Ite,unsigned int i, 
+  template <typename Map_,typename Ite,unsigned int i,
             unsigned int dim,bool Const>
   class CMap_cell_iterator<Map_,Ite,i,dim,Const,Tag_true>: public Ite
   {
@@ -64,12 +64,12 @@ namespace CGAL {
     void unmark_treated_darts()
     {
       if (this->mmap->is_whole_map_unmarked(mcell_mark_number)) return;
-            
+
       this->mmap->negate_mark(this->mmark_number);
       this->mmap->negate_mark(mcell_mark_number);
 
       Ite::rewind();
-      mark_cell<Map,i,dim>(*this->mmap, (*this), 
+      mark_cell<Map,i,dim>(*this->mmap, (*this),
                            mcell_mark_number);
 
       while (this->mmap->number_of_unmarked_darts(mcell_mark_number) > 0 ||
@@ -77,14 +77,14 @@ namespace CGAL {
       {
         this->operator++();
       }
-      
+
       this->mmap->negate_mark(mcell_mark_number);
       this->mmap->negate_mark(this->mmark_number);
 
       CGAL_assertion(this->mmap->is_whole_map_unmarked(this->mmark_number));
       CGAL_assertion(this->mmap->is_whole_map_unmarked(mcell_mark_number));
     }
-      
+
   public:
     /// Main constructor.
     CMap_cell_iterator(Map& amap, Dart_handle adart):
@@ -94,10 +94,10 @@ namespace CGAL {
       CGAL_static_assertion( (boost::is_same<typename Ite::Basic_iterator,
                               Tag_true>::value) );
       CGAL_assertion(amap.is_whole_map_unmarked(mcell_mark_number));
-      
+
       mark_cell<Map,i,dim>(amap, adart, mcell_mark_number);
     }
-  
+
     /// Destructor.
     ~CMap_cell_iterator()
     {
@@ -105,6 +105,7 @@ namespace CGAL {
         unmark_treated_darts();
       this->mmap->free_mark(mcell_mark_number);
       this->mmap->free_mark(this->mmark_number);
+      this->mcell_mark_number = -1; // To avoid basic class to try to unmark darts.
       this->mmark_number = -1; // To avoid basic class to try to unmark darts.
     }
 
@@ -147,10 +148,10 @@ namespace CGAL {
       }
       while (this->cont() &&
              this->mmap->is_marked((*this), mcell_mark_number));
-      
+
       if (this->cont())
       {
-        mark_cell<Map,i,dim>(*this->mmap, (*this), 
+        mark_cell<Map,i,dim>(*this->mmap, (*this),
                              mcell_mark_number);
       }
       return *this;
@@ -168,7 +169,7 @@ namespace CGAL {
   /* Class CMap_cell_iterator<Map,Ite,i,dim,Tag_false>: to iterate onto the
    * cells incident to the given iterator which does not use mark.
    */
-  template <typename Map_,typename Ite, unsigned int i, 
+  template <typename Map_,typename Ite, unsigned int i,
             unsigned int dim,bool Const>
   class CMap_cell_iterator<Map_,Ite,i,dim,Const,Tag_false>: public Ite
   {
@@ -190,7 +191,7 @@ namespace CGAL {
       if (this->mmap->is_whole_map_unmarked(mmark_number)) return;
 
       Ite::rewind();
-      mark_cell<Map,i,dim>(*this->mmap, (*this), 
+      mark_cell<Map,i,dim>(*this->mmap, (*this),
                            mmark_number);
       while (this->mmap->number_of_unmarked_darts(mmark_number) > 0)
         this->operator++();
@@ -205,7 +206,7 @@ namespace CGAL {
       mmark_number(amap.get_new_mark())
     {
       CGAL_static_assertion( (boost::is_same<typename Ite::Basic_iterator,
-                              Tag_true>::value) );      
+                              Tag_true>::value) );
       CGAL_assertion(amap.is_whole_map_unmarked(mmark_number));
       mark_cell<Map,i,dim>(amap, adart, mmark_number);
     }
@@ -216,6 +217,7 @@ namespace CGAL {
       if (this->mmap->get_number_of_times_mark_reserved(mmark_number)==1)
         unmark_treated_darts();
       this->mmap->free_mark(mmark_number);
+      this->mmark_number = -1; // To avoid basic class to try to unmark darts.
     }
 
     /// Copy constructor.
@@ -270,12 +272,12 @@ namespace CGAL {
   };
   //****************************************************************************
   /* Class CMap_cell_iterator<Map,CMap_dart_iterator_basic_of_all<Map>,
-     i,dim,Tag_false>: specialization to iterate onto 
+     i,dim,Tag_false>: specialization to iterate onto
      * all the cells of the map.
      */
   template <typename Map_,unsigned int i,unsigned int dim,bool Const>
   class CMap_cell_iterator<Map_,CMap_dart_iterator_basic_of_all<Map_,Const>,
-                           i,dim,Const,Tag_false>: 
+                           i,dim,Const,Tag_false>:
     public CMap_dart_iterator_basic_of_all<Map_,Const>
   {
   public:
@@ -284,7 +286,7 @@ namespace CGAL {
 
     typedef typename Base::Dart_handle Dart_handle;
     typedef typename Base::Map Map;
-  
+
   protected:
     /// Unmark all the marked darts during the iterator.
     void unmark_treated_darts()
@@ -296,7 +298,7 @@ namespace CGAL {
       if (this->mmap->is_whole_map_unmarked(mmark_number)) return;
 
       Base::rewind();
-      mark_cell<Map,i,dim>(*this->mmap, (*this), 
+      mark_cell<Map,i,dim>(*this->mmap, (*this),
                            mmark_number);
       while (this->mmap->number_of_unmarked_darts(mmark_number) > 0)
         this->operator++();
@@ -316,12 +318,22 @@ namespace CGAL {
       mark_cell<Map,i,dim>(amap, (*this), mmark_number);
     }
 
+   /// Constructor with a dart in parameter (for end iterator).
+    CMap_cell_iterator(Map& amap, Dart_handle adart):
+      Base(amap, adart),
+      mmark_number(amap.get_new_mark())
+    {
+      if (adart!=this->mmap->null_handle)
+        mark_cell<Map,i,dim>(amap, (*this), mmark_number);
+    }
+
     /// Destructor.
     ~CMap_cell_iterator()
     {
       if (this->mmap->get_number_of_times_mark_reserved(mmark_number)==1)
         unmark_treated_darts();
       this->mmap->free_mark(mmark_number);
+      this->mmark_number = -1; // To avoid basic class to try to unmark darts.
     }
 
     /// Copy constructor.
@@ -378,10 +390,10 @@ namespace CGAL {
   /* Class CMap_one_dart_per_incident_cell_iterator<Map,i,j,dim>: to iterate
    * onto one dart per i-cell incident to the given j-cell.
    */
-  template <typename Map_,unsigned int i,unsigned int j, 
+  template <typename Map_,unsigned int i,unsigned int j,
             unsigned int dim=Map_::dimension,bool Const=false>
   class CMap_one_dart_per_incident_cell_iterator:
-    public CMap_cell_iterator<Map_, 
+    public CMap_cell_iterator<Map_,
                               CMap_dart_iterator_basic_of_cell
                               <Map_,j,dim,Const>, i,dim,Const>
   {
@@ -394,12 +406,12 @@ namespace CGAL {
 
     typedef typename Base::Dart_handle Dart_handle;
     typedef typename Base::Map Map;
-    
+
     typedef Tag_false Use_mark;
     typedef Tag_false Basic_iterator;
-    
+
     /// Main constructor.
-    CMap_one_dart_per_incident_cell_iterator(Map& amap, Dart_handle adart): 
+    CMap_one_dart_per_incident_cell_iterator(Map& amap, Dart_handle adart):
       Base(amap, adart)
     {}
   };
@@ -416,21 +428,22 @@ namespace CGAL {
   public:
     typedef CMap_one_dart_per_cell_iterator<Map_,i,dim,Const> Self;
     typedef CMap_cell_iterator<Map_,
-                               CMap_dart_iterator_basic_of_all<Map_,Const>, 
+                               CMap_dart_iterator_basic_of_all<Map_,Const>,
                                i,dim,Const> Base;
 
     typedef typename Base::Dart_handle Dart_handle;
     typedef typename Base::Map Map;
-    
+
     typedef Tag_false Use_mark;
     typedef Tag_false Basic_iterator;
-    
+
     /// Main constructor.
     CMap_one_dart_per_cell_iterator(Map& amap): Base(amap)
     {}
     /// Constructor with a dart in parameter (for end iterator).
-    CMap_one_dart_per_cell_iterator(Map& amap, Dart_handle adart): Base(amap)
-    { this->set_current_dart(adart); }
+    CMap_one_dart_per_cell_iterator(Map& amap, Dart_handle adart):
+	     Base(amap, adart)
+    {}
   };
 //****************************************************************************
 //****************************************************************************
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map.h b/Combinatorial_map/include/CGAL/Combinatorial_map.h
index cf4a06b7815..2759b214dbe 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map.h
@@ -3265,7 +3265,7 @@ namespace CGAL {
       iterator end()   { return iterator(mmap,mmap.null_handle); }
       const_iterator begin() const { return const_iterator(mmap); }
       const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
-      size_type size()
+      size_type size() const
       { return mmap.number_of_darts(); }
       bool empty() const
       { return mmap.is_empty(); }
@@ -3337,10 +3337,10 @@ namespace CGAL {
       iterator end()   { return iterator(mmap,mmap.null_handle); }
       const_iterator begin() const { return const_iterator(mmap); }
       const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
-      size_type size()
+      size_type size() const
       {
         if (msize==0)
-          for ( const_iterator it=begin(); it!=end(); ++it)
+          for ( const_iterator it=begin(), itend=end(); it!=end(); ++it)
             ++msize;
         return msize;
       }
@@ -3348,7 +3348,7 @@ namespace CGAL {
       { return mmap.is_empty(); }
     private:
       Self & mmap;
-      size_type msize;
+      mutable size_type msize;
     };
     //**************************************************************************
     // One_dart_per_cell_const_range
@@ -3360,10 +3360,10 @@ namespace CGAL {
       {}
       const_iterator begin() const { return const_iterator(mmap); }
       const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
-      size_type size()
+      size_type size() const
       {
         if (msize==0)
-          for ( const_iterator it=begin(); it!=end(); ++it)
+          for ( const_iterator it=begin(), itend=end(); it!=end(); ++it)
             ++msize;
         return msize;
       }
@@ -3371,7 +3371,7 @@ namespace CGAL {
       { return mmap.is_empty(); }
     private:
       const Self & mmap;
-      size_type msize;
+      mutable size_type msize;
     };
     //**************************************************************************
     /// @return a range on the i-cells incindent to the given j-cell.
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_iterators_base.h b/Combinatorial_map/include/CGAL/Combinatorial_map_iterators_base.h
index 5f94af54381..8ae269c562f 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_iterators_base.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_iterators_base.h
@@ -36,6 +36,10 @@ namespace CGAL {
    *    the involution Bi.
    *  - CMap_non_basic_iterator<Map_,Ite> to transform the basic iterator Ite
    *    into the corresponding non basic iterator.
+   *  - CMap_range<Map,It,ConstIt,BasicIt> generic definition of a range
+   *    given an iterator and its const version
+   *  - CMap_const_range<Map,It,ConstIt,BasicIt> generic definition of a const
+   *    range given an iterator and its const version
    */
   //****************************************************************************
   /// OperationState: type to keep the last operation used by the previous ++.
@@ -56,7 +60,7 @@ namespace CGAL {
     OP_BETA21,     ///< Previous op was beta21.
     OP_JUMP,      ///< Previous op was a jump .
     OP_POP,       ///< Previous op pop a dart from a stack or a queue.
-    OP_END        ///< Previous op go out of the iterator.    
+    OP_END        ///< Previous op go out of the iterator.
   };
   //****************************************************************************
   /** Generic class of iterator onto darts.
@@ -87,14 +91,14 @@ namespace CGAL {
     typedef typename Base::value_type value_type;
     typedef typename Base::difference_type difference_type;
     typedef typename Base::pointer pointer;
-    typedef typename Base::reference reference;    
+    typedef typename Base::reference reference;
 
     /// true iff this iterator is basic
     typedef Tag_true Basic_iterator;
 
   public:
     /// Main constructor.
-    CMap_dart_iterator(Map& amap, Dart_handle adart): 
+    CMap_dart_iterator(Map& amap, Dart_handle adart):
       Base(adart),
       mmap(&amap),
       mfirst_dart(adart),
@@ -105,7 +109,7 @@ namespace CGAL {
     bool operator==(const Self& aiterator) const
     {
       return ( ((*this==mmap->null_handle) && (aiterator==mmap->null_handle)) ||
-               (mfirst_dart == aiterator.mfirst_dart && 
+               (mfirst_dart == aiterator.mfirst_dart &&
                static_cast<const Base&>(*this)==
                 static_cast<const Base&>(aiterator)) );
     }
@@ -119,7 +123,7 @@ namespace CGAL {
 
     /// Accessor to the combinatorial map.
     Map* get_combinatorial_map() const { return mmap; }
-    
+
     /// Rewind of the iterator to its beginning.
     void rewind()
     { set_current_dart(mfirst_dart); mprev_op = OP_NONE; }
@@ -179,7 +183,7 @@ namespace CGAL {
     OperationState mprev_op;
   };
   //****************************************************************************
-  /* Class CMap_extend_iterator<Map,Ite,Bi> which extend a given iterator by 
+  /* Class CMap_extend_iterator<Map,Ite,Bi> which extend a given iterator by
    * adding Bi and by using a stack and a mark.
    * General case when Ite does not have already a stack.
    */
@@ -198,7 +202,7 @@ namespace CGAL {
 
     CGAL_static_assertion( (Bi<=Map::dimension &&
                             boost::is_same<Ite_has_stack,Tag_false>::value) );
-    
+
   public:
     /// Main constructor.
     CMap_extend_iterator(Map& amap, Dart_handle adart, int amark):
@@ -247,7 +251,7 @@ namespace CGAL {
       }
       while ( this->cont() &&
               this->mmap->is_marked(*this, mmark_number) );
-      
+
       if ( !this->cont() )
       {
         if ( !mto_treat.empty() )
@@ -300,7 +304,7 @@ namespace CGAL {
     Dart_handle minitial_dart;
   };
   //****************************************************************************
-  /* Class CMap_extend_iterator<Map,Ite,Bi> which extend a given iterator by 
+  /* Class CMap_extend_iterator<Map,Ite,Bi> which extend a given iterator by
    * adding Bi and by using a stack and a mark.
    * Specialization when Ite has already a stack.
    */
@@ -371,18 +375,18 @@ namespace CGAL {
   //****************************************************************************
   //* Class CMap_non_basic_iterator allows to transform a basic_iterator onto
   //* a non basic one, depending if the basic iterator uses mark or not.
-  template <typename Map_,typename Basic_iterator, 
+  template <typename Map_,typename Basic_iterator,
             typename Use_mark=typename Basic_iterator::Use_mark>
   class CMap_non_basic_iterator;
   //****************************************************************************
   template <typename Map_,typename Base_>
-  class CMap_non_basic_iterator<Map_,Base_,Tag_true>: 
+  class CMap_non_basic_iterator<Map_,Base_,Tag_true>:
     public Base_
   {
   public:
     typedef CMap_non_basic_iterator<Map_,Base_,Tag_true> Self;
     typedef Base_ Base;
-    
+
     typedef typename Base::Map Map;
     typedef typename Base::Dart_handle Dart_handle;
 
@@ -391,7 +395,7 @@ namespace CGAL {
 
     CGAL_static_assertion( (boost::is_same<typename Base::Basic_iterator,
                             Tag_true>::value) );
-    
+
     /// Main constructor.
     CMap_non_basic_iterator(Map& amap, Dart_handle adart1):
       Base(amap, adart1, amap.get_new_mark())
@@ -405,11 +409,12 @@ namespace CGAL {
           (this->mmark_number)==1)
         unmark_treated_darts();
       this->mmap->free_mark(this->mmark_number);
+      this->mmark_number = -1; // To avoid basic class to try to unmark darts.
     }
 
     /// Copy constructor.
     CMap_non_basic_iterator(const Self& aiterator):
-      Base(aiterator)      
+      Base(aiterator)
     { this->mmap->share_a_mark(this->mmark_number); }
 
     /// Assignment operator.
@@ -421,6 +426,7 @@ namespace CGAL {
             (this->mmark_number)==1)
           unmark_treated_darts();
         this->mmap->free_mark(this->mmark_number);
+        this->mmark_number = -1;
 
         Base::operator=(aiterator);
         this->mmap->share_a_mark(this->mmark_number);
@@ -460,13 +466,13 @@ namespace CGAL {
   };
   //****************************************************************************
   template <typename Map_,typename Base_>
-  class CMap_non_basic_iterator<Map_,Base_,Tag_false>: 
+  class CMap_non_basic_iterator<Map_,Base_,Tag_false>:
     public Base_
   {
   public:
     typedef CMap_non_basic_iterator<Map_,Base_,Tag_false> Self;
     typedef Base_ Base;
-    
+
     typedef typename Base::Map Map;
     typedef typename Base::Dart_handle Dart_handle;
 
@@ -475,7 +481,7 @@ namespace CGAL {
 
     CGAL_static_assertion( (boost::is_same<typename Base::Basic_iterator,
                             Tag_true>::value) );
-    
+
     /// Main constructor.
     CMap_non_basic_iterator(Map& amap, Dart_handle adart):
       Base(amap, adart)
@@ -495,10 +501,10 @@ namespace CGAL {
     iterator end()               { return iterator(mmap,mmap.null_handle); }
     const_iterator begin() const { return const_iterator(mmap,mdart); }
     const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
-    typename Map_::size_type size()
+    typename Map_::size_type size() const
     {
       if (msize==0)
-        for (const_iterator it=begin(); it!=end(); ++it)
+        for ( const_iterator it=begin(), itend=end(); it!=end(); ++it)
           ++msize;
       return msize;
     }
@@ -507,7 +513,7 @@ namespace CGAL {
   private:
     Map_ & mmap;
     typename Map_::Dart_handle mdart;
-    typename Map_::size_type msize;
+    mutable typename Map_::size_type msize;
   };
   //****************************************************************************
   template <typename Map_, typename It, typename Const_it>
@@ -523,7 +529,7 @@ namespace CGAL {
     iterator end()               { return iterator(mmap,mmap.null_handle,mmark); }
     const_iterator begin() const { return const_iterator(mmap,mdart,mmark); }
     const_iterator end() const   { return const_iterator(mmap,mmap.null_handle,mmark); }
-    typename Map_::size_type size()
+    typename Map_::size_type size() const
     {
       if (msize==0)
         for ( CMap_non_basic_iterator<Map_,const_iterator> it(mmap,mdart);
@@ -536,7 +542,7 @@ namespace CGAL {
   private:
     Map_ & mmap;
     typename Map_::Dart_handle mdart;
-    typename Map_::size_type msize;
+    mutable typename Map_::size_type msize;
     int mmark;
   };
   //****************************************************************************
@@ -550,10 +556,10 @@ namespace CGAL {
     {}
     const_iterator begin() const { return const_iterator(mmap,mdart); }
     const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
-    typename Map_::size_type size()
+    typename Map_::size_type size() const
     {
       if (msize==0)
-        for (const_iterator it=begin(); it!=end(); ++it)
+        for ( const_iterator it=begin(), itend=end(); it!=end(); ++it)
           ++msize;
       return msize;
     }
@@ -562,7 +568,7 @@ namespace CGAL {
   private:
     const Map_ & mmap;
     typename Map_::Dart_const_handle mdart;
-    typename Map_::size_type msize;
+    mutable typename Map_::size_type msize;
   };
   //****************************************************************************
   template <typename Map_, typename Const_it>
@@ -575,7 +581,7 @@ namespace CGAL {
     {}
     const_iterator begin() const { return const_iterator(mmap,mdart,mmark); }
     const_iterator end() const   { return const_iterator(mmap,mmap.null_handle,mmark); }
-    typename Map_::size_type size()
+    typename Map_::size_type size() const
     {
       if (msize==0)
         for ( CMap_non_basic_iterator<Map_,const_iterator> it(mmap,mdart);
@@ -588,7 +594,7 @@ namespace CGAL {
   private:
     const Map_ & mmap;
     typename Map_::Dart_const_handle mdart;
-    typename Map_::size_type msize;
+    mutable typename Map_::size_type msize;
     int mmark;
   };
   //****************************************************************************
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex_constructors.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex_constructors.h
index 68a8ce83ce2..e426b3453f7 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex_constructors.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex_constructors.h
@@ -545,6 +545,8 @@ namespace CGAL {
                                     <typename LCC::Traits> > (alcc, P);
   }
 
+  /** Export the alcc in off file format. If dimension>2, export all faces but only once.
+   */
   template < class LCC >
   void write_off(LCC& alcc, std::ostream& out)
   {
@@ -559,7 +561,7 @@ namespace CGAL {
     writer.write_header( out,
                          alcc.number_of_vertex_attributes(),
                          alcc.number_of_darts(),
-                         alcc.template one_dart_per_cell<2,2>().size() );
+                         alcc.template one_dart_per_cell<2>().size() );
 
     typedef typename LCC::Vertex_attribute_range::iterator VCI;
     VCI vit, vend = alcc.vertex_attributes().end();
@@ -584,23 +586,28 @@ namespace CGAL {
       {
         std::size_t n = 0;
         // First we count the number of vertices of the face.
-        for ( typename LCC::template Dart_of_cell_range<2,2>::iterator
-                itf=alcc.template darts_of_cell<2,2>(itall).begin(),
-                itfend=alcc.template darts_of_cell<2,2>(itall).end();
+        for ( typename LCC::template Dart_of_orbit_range<1>::iterator
+                itf=alcc.template darts_of_orbit<1>(itall).begin(),
+                itfend=alcc.template darts_of_orbit<1>(itall).end();
               itf!=itfend; ++itf, ++n );
 
         CGAL_assertion( n>=3 );
         writer.write_facet_begin(n);
 
         // Second we write the indices of vertices.
-        for ( typename LCC::template Dart_of_cell_range<2,2>::iterator
-                itf=alcc.template darts_of_cell<2,2>(itall).begin(),
-                itfend=alcc.template darts_of_cell<2,2>(itall).end();
+        for ( typename LCC::template Dart_of_orbit_range<1>::iterator
+                itf=alcc.template darts_of_orbit<1>(itall).begin(),
+                itfend=alcc.template darts_of_orbit<1>(itall).end();
               itf!=itfend; ++itf )
         {
           // TODO case with index
           writer.write_facet_vertex_index(index[VCI(alcc.vertex_attribute(itf))]);
-          alcc.mark(itf, m);
+
+          for ( typename LCC::template Dart_of_involution_basic_range<1>::iterator
+                  itinv=alcc.template darts_of_involution_basic<1>(itf, m).begin(),
+                  itinvend=alcc.template darts_of_involution_basic<1>(itf, m).end();
+                itinv!=itinvend; ++itinv )
+            alcc.mark(itinv, m);
         }
         writer.write_facet_end();
       }
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex_storages.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex_storages.h
index 148d383d8cb..29243b4b017 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex_storages.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex_storages.h
@@ -90,6 +90,15 @@ namespace CGAL {
         public Helper::template Attribute_const_range<i>
     {};
 
+    typedef typename Attribute_type<0>::type Vertex_attribute;
+    typedef typename Attribute_handle<0>::type Vertex_attribute_handle;
+    typedef typename Attribute_const_handle<0>::type
+    Vertex_attribute_const_handle;
+
+    typedef typename Attribute_range<0>::type Vertex_attribute_range;
+    typedef typename Attribute_const_range<0>::type
+    Vertex_attribute_const_range;
+
     /// Number of marks
     static const size_type NB_MARKS = 32;