diff --git a/BGL/include/CGAL/boost/graph/graph_traits_Linear_cell_complex.h b/BGL/include/CGAL/boost/graph/graph_traits_Linear_cell_complex.h
index 388c8424613..854e24b73c6 100644
--- a/BGL/include/CGAL/boost/graph/graph_traits_Linear_cell_complex.h
+++ b/BGL/include/CGAL/boost/graph/graph_traits_Linear_cell_complex.h
@@ -203,9 +203,9 @@ typename boost::graph_traits<CGAL_LCC_TYPE >::vertex_descriptor
 source(typename boost::graph_traits<CGAL_LCC_TYPE >::edge_descriptor e,
        const CGAL_LCC_TYPE& amap)
 {
-  return e->template attribute<0>();
-  /* return const_cast<CGAL_LCC_TYPE&>(amap).template beta<2>(e)->
-      template attribute<0>(); */
+  // return e->template attribute<0>();
+  return const_cast<CGAL_LCC_TYPE&>(amap).template beta<2>(e)->
+      template attribute<0>();
 }
 
 CGAL_LCC_TEMPLATE_ARGS
@@ -213,9 +213,9 @@ typename boost::graph_traits<CGAL_LCC_TYPE >::vertex_descriptor
 target(typename boost::graph_traits<CGAL_LCC_TYPE >::edge_descriptor e,
        const CGAL_LCC_TYPE& amap)
 {
-  // return e->template attribute<0>();
-  return const_cast<CGAL_LCC_TYPE&>(amap).template beta<2>(e)->
-        template attribute<0>();
+  return e->template attribute<0>();
+  /*return const_cast<CGAL_LCC_TYPE&>(amap).template beta<2>(e)->
+        template attribute<0>();*/
 }
 
 CGAL_LCC_TEMPLATE_ARGS
@@ -271,8 +271,8 @@ degree(typename boost::graph_traits<CGAL_LCC_TYPE >::vertex_descriptor v,
   typename boost::graph_traits<CGAL_LCC_TYPE >::degree_size_type degree=0;
 
   for (typename CGAL_LCC_TYPE::template Dart_of_cell_range<0>::const_iterator
-         it=cm.template darts_of_cell<0>(halfedge(v, cm)).begin(),
-         itend=cm.template darts_of_cell<0>(halfedge(v, cm)).end();
+         it=cm.template darts_of_cell<0>(const_cast<CGAL_LCC_TYPE&>(cm).template beta<2>(v->dart())).begin(),
+         itend=cm.template darts_of_cell<0>(const_cast<CGAL_LCC_TYPE&>(cm).template beta<2>(v->dart())).end();
        it!=itend; ++it)
   {
     ++degree;
@@ -361,8 +361,8 @@ void remove_edge(typename boost::graph_traits<CGAL_LCC_TYPE>::vertex_descriptor
   if ( e.second )
   {
     assert ( !cm.template is_free<2>(e.first.first_halfedge()) );
-    cm.erase_dart(cm.template beta<2>(e.first.first_halfedge()));
-    cm.erase_dart(e.first.first_halfedge());
+    cm.restricted_erase_dart(cm.template beta<2>(e.first.first_halfedge()));
+    cm.restricted_erase_dart(e.first.first_halfedge());
   }
 }
 
@@ -372,8 +372,8 @@ remove_edge(typename boost::graph_traits<CGAL_LCC_TYPE>::edge_descriptor e,
             CGAL_LCC_TYPE& cm)
 {
   assert ( !cm.template is_free<2>(e.first_halfedge()) );
-  cm.erase_dart(cm.template beta<2>(e.first_halfedge()));
-  cm.erase_dart(e);
+  cm.restricted_erase_dart(cm.template beta<2>(e.first_halfedge()));
+  cm.restricted_erase_dart(e);
 }
 
 CGAL_LCC_TEMPLATE_ARGS
@@ -396,7 +396,7 @@ void
 remove_vertex(typename boost::graph_traits<CGAL_LCC_TYPE>::vertex_descriptor v, 
               CGAL_LCC_TYPE& cm)
 {
-  // cm.template erase_attribute<0>(v);
+  cm.template erase_attribute<0>(v);
   // Useled because in CMap, attributes are automatically deleted thanks
   // to ref counting
 }
@@ -433,11 +433,13 @@ halfedge(typename boost::graph_traits<CGAL_LCC_TYPE>::vertex_descriptor u,
          const CGAL_LCC_TYPE& g) 
 {
   for (typename CGAL_LCC_TYPE::template Dart_of_cell_range<0>::iterator
-         it=const_cast<CGAL_LCC_TYPE&>(g).template darts_of_cell<0>(u->dart()).begin(),
-         itend=const_cast<CGAL_LCC_TYPE&>(g).template darts_of_cell<0>(u->dart()).end();
+         it=const_cast<CGAL_LCC_TYPE&>(g).template
+       darts_of_cell<0>(const_cast<CGAL_LCC_TYPE&>(g).template beta<2>(u->dart())).begin(),
+         itend=const_cast<CGAL_LCC_TYPE&>(g).template
+       darts_of_cell<0>(const_cast<CGAL_LCC_TYPE&>(g).template beta<2>(u->dart())).end();
        it!=itend; ++it)
   {
-    if (g.template beta<2>(it)->template attribute<0>()==v)
+    if (it->template attribute<0>()==v)
     {
       return std::make_pair(it, true);
       // return std::make_pair(const_cast<CGAL_LCC_TYPE&>(g).template beta<2>(it), true);
@@ -495,8 +497,11 @@ add_face(InputIterator begin, InputIterator end, CGAL_LCC_TYPE& cm)
 
 CGAL_LCC_TEMPLATE_ARGS
 bool is_valid(const CGAL_LCC_TYPE& cm, bool = false)
-{ return cm.is_valid(); }
-  
+{
+  // cm.display_darts(std::cout,true);
+  return cm.is_valid(true); // true to inverse the convention between darts and 0-attributes
+}
+
 CGAL_LCC_TEMPLATE_ARGS
 Iterator_range<typename boost::graph_traits<CGAL_LCC_TYPE>::halfedge_iterator>
 halfedges(const CGAL_LCC_TYPE& cm)
@@ -525,7 +530,8 @@ void set_target(typename boost::graph_traits<CGAL_LCC_TYPE>::halfedge_descriptor
                 typename boost::graph_traits<CGAL_LCC_TYPE>::vertex_descriptor v,
                 CGAL_LCC_TYPE& cm)
 {
-  cm.template set_dart_attribute<0>(cm.template beta<2>(h1), v);
+  // cm.template set_dart_attribute<0>(cm.template beta<2>(h1), v);
+  cm.template restricted_set_dart_attribute<0>(h1, v);
   // cm.template set_dart_attribute<0>(h1, v);
 }
 
@@ -540,8 +546,8 @@ void set_halfedge(typename boost::graph_traits<CGAL_LCC_TYPE>::vertex_descriptor
                   typename boost::graph_traits<CGAL_LCC_TYPE>::halfedge_descriptor h,
                   CGAL_LCC_TYPE& cm)
 {
-  v->set_dart(cm.template beta<2>(h));
-  // v->set_dart(h);
+  // v->set_dart(cm.template beta<2>(h));
+  v->set_dart(h);
 }
 
 CGAL_LCC_TEMPLATE_ARGS
@@ -553,7 +559,7 @@ CGAL_LCC_TEMPLATE_ARGS
 void remove_face(typename boost::graph_traits<CGAL_LCC_TYPE>::face_descriptor f,
                  CGAL_LCC_TYPE& cm)
 {
-  // cm.template erase_attribute<2>(f);
+  cm.template erase_attribute<2>(f);
   // Useled because in CMap, attributes are automatically deleted thanks
   // to ref counting
 }
@@ -562,7 +568,10 @@ CGAL_LCC_TEMPLATE_ARGS
 void set_face(typename boost::graph_traits<CGAL_LCC_TYPE>::halfedge_descriptor h,              
               typename boost::graph_traits<CGAL_LCC_TYPE>::face_descriptor f,
               CGAL_LCC_TYPE& cm)
-{ cm.template set_dart_attribute<2>(h, f); }
+{
+  cm.template restricted_set_dart_attribute<2>(h, f);
+  // cm.template set_dart_attribute<2>(h, f);
+}
 
 CGAL_LCC_TEMPLATE_ARGS
 void set_halfedge(typename boost::graph_traits<CGAL_LCC_TYPE>::face_descriptor f,
diff --git a/Combinatorial_map/include/CGAL/Cell_attribute.h b/Combinatorial_map/include/CGAL/Cell_attribute.h
index 245e9a07d09..17ee349e1f2 100644
--- a/Combinatorial_map/include/CGAL/Cell_attribute.h
+++ b/Combinatorial_map/include/CGAL/Cell_attribute.h
@@ -85,6 +85,9 @@ namespace CGAL {
     template<typename, unsigned int, typename>
     friend struct internal::Decrease_attribute_functor_run;
 
+    template<typename, unsigned int, typename>
+    friend struct internal::Restricted_decrease_attribute_functor_run;
+
     template <typename, typename>
     friend struct internal::Reverse_orientation_of_map_functor;
 
@@ -197,6 +200,9 @@ namespace CGAL {
     template<typename, unsigned int, typename>
     friend struct internal::Decrease_attribute_functor_run;
 
+    template<typename, unsigned int, typename>
+    friend struct internal::Restricted_decrease_attribute_functor_run;
+
     template <typename, typename>
     friend struct internal::Reverse_orientation_of_map_functor;
 
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map.h b/Combinatorial_map/include/CGAL/Combinatorial_map.h
index 2c428d342af..121ea6d35e9 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map.h
@@ -490,6 +490,27 @@ namespace CGAL {
       mdarts.erase(adart);
     }
 
+    /** Erase a dart from the list of darts. Restricted version
+     *  which do not delete attribute having no more dart associated.
+     * @param adart the dart to erase.
+     */
+    void restricted_erase_dart(Dart_handle adart)
+    {
+      // 1) We update the number of marked darts.
+      for ( size_type i = 0; i < mnb_used_marks; ++i)
+      {
+        if (is_marked(adart, mused_marks_stack[i]))
+          --mnb_marked_darts[mused_marks_stack[i]];
+      }
+
+      // 2) We update the attribute_ref_counting.
+      Helper::template Foreach_enabled_attributes
+        <internal::Restricted_decrease_attribute_functor<Self> >::run(this,adart);
+
+      // 3) We erase the dart.
+      mdarts.erase(adart);
+    }
+
     /// @return true if dh points to a used dart (i.e. valid).
     bool is_dart_used(Dart_const_handle dh) const
     { return mdarts.is_used(dh); }
@@ -565,6 +586,32 @@ namespace CGAL {
       return null_handle;
     }
 
+    // Set the handle on the i th attribute
+    // Restricted version which do not use delete attributes when their ref
+    // counting become null, nor that update the dart of attribute.
+    template<unsigned int i>
+    void restricted_set_dart_attribute(Dart_handle dh,
+                                       typename Attribute_handle<i>::type ah)
+    {
+      CGAL_static_assertion_msg(Helper::template Dimension_index<i>::value>=0,
+                     "set_dart_attribute<i> called but i-attributes are disabled.");
+
+      if ( this->template attribute<i>(dh)==ah ) return;
+
+      if ( this->template attribute<i>(dh)!=null_handle )
+      {
+        this->template get_attribute<i>(this->template attribute<i>(dh)).
+          dec_nb_refs();
+      }
+
+      Base::template basic_set_dart_attribute<i>(dh, ah);
+
+      if ( ah!=null_handle )
+      {
+        this->template get_attribute<i>(ah).inc_nb_refs();
+      }
+    }
+
     // Set the handle on the i th attribute
     template<unsigned int i>
     void set_dart_attribute(Dart_handle dh,
@@ -1093,9 +1140,13 @@ namespace CGAL {
     }
 
     /** Test if the map is valid.
+     * @param reverseextremity to inverse the convention between source and
+     *        target of a dart. With false (default), a dart is associated with
+     *        a 0-attribute for its source (origin);
+     *        with true this is for its target (as in hds or surface mesh).
      * @return true iff the map is valid.
      */
-    bool is_valid() const
+    bool is_valid(bool reverseextremity=false) const
     {
       bool valid = true;
       unsigned int i = 0, j = 0;
@@ -1113,7 +1164,10 @@ namespace CGAL {
         if ( !valid )
         { // We continue the traversal to mark all the darts.
           for ( i=0; i<=dimension; ++i)
-            if (marks[i]!=INVALID_MARK) mark(it,marks[i]);
+            if (marks[i]!=INVALID_MARK)
+            {
+              mark(it,marks[i]);
+            }
         }
         else
         {
@@ -1182,7 +1236,7 @@ namespace CGAL {
           }
           Helper::template Foreach_enabled_attributes
             <internal::Test_is_valid_attribute_functor<Self> >::
-            run(this,it,&marks,&valid);
+            run(this,it,&marks,&valid, reverseextremity);
         }
       }
       for ( i=0; i<=dimension; ++i)
@@ -1247,7 +1301,7 @@ namespace CGAL {
         for ( unsigned int i=0; i<=dimension; ++i)
         {
           os << &(*it->beta(i)) << ",\t";
-          if (it->is_free(i)) os << "\t";
+          if (is_free(it, i)) os << "\t";
         }
         if ( attribs )
         {
diff --git a/Combinatorial_map/include/CGAL/internal/Combinatorial_map_internal_functors.h b/Combinatorial_map/include/CGAL/internal/Combinatorial_map_internal_functors.h
index a399f0e75d6..6b18569529f 100644
--- a/Combinatorial_map/include/CGAL/internal/Combinatorial_map_internal_functors.h
+++ b/Combinatorial_map/include/CGAL/internal/Combinatorial_map_internal_functors.h
@@ -52,6 +52,10 @@
  * internal::Decrease_attribute_functor<CMap> to decrease by one the ref
  *    counting of a given i-attribute.
  *
+ * internal::Restricted_decrease_attribute_functor<CMap> to decrease by one the
+ *    ref counting of a given i-attribute, but without deleting attribute
+ *    having nomore dart associated with.
+ *
  * internal::Beta_functor<Dart, i...> to call several beta on the given dart.
  *   Indices are given as parameter of the run function.
  *
@@ -184,6 +188,10 @@ struct Test_is_valid_attribute_functor
    * ie all the darts belonging to a i-cell are linked to the same attribute.
    * @param adart a dart.
    * @param amark a mark used to mark darts of the i-cell.
+   * @param reverseextremity to inverse the convention between source and
+   *        target of a dart. With false (default), a dart is associated with
+   *        a 0-attribute for its source (origin);
+   *        with true this is for its target (as in hds or surface mesh).
    * @return true iff all the darts of the i-cell link to the same attribute.
    */
   typedef typename CMap::size_type size_type;
@@ -191,7 +199,8 @@ struct Test_is_valid_attribute_functor
   template <unsigned int i>
   static void run(const CMap* amap,
                   typename CMap::Dart_const_handle adart,
-                  std::vector<size_type>* marks, bool *ares)
+                  std::vector<size_type>* marks, bool *ares,
+                  bool reverseextremity=false)
   {
     CGAL_static_assertion_msg(CMap::Helper::template
                               Dimension_index<i>::value>=0,
@@ -207,25 +216,42 @@ struct Test_is_valid_attribute_functor
     typename CMap::template Attribute_const_handle<i>::type
         a=amap->template attribute<i>(adart);
 
+    if (i==0 && reverseextremity)
+    {
+      a=amap->template attribute<i>(amap->template beta<2>(adart));
+    }
+
     unsigned int nb = 0;
     for ( typename
             CMap::template Dart_of_cell_basic_const_range<i>::const_iterator
             it=amap->template darts_of_cell_basic<i>(adart, amark).begin();
           it.cont(); ++it )
     {
-      if ( amap->template attribute<i>(it) != a )
+      if ( (i==0 && reverseextremity &&
+            amap->template attribute<i>(amap->template beta<2>(it))!= a) ||
+           ((i>0 || !reverseextremity) && amap->template attribute<i>(it) != a) )
       {
         std::cout<<"ERROR: an attribute of the "<<i<<"-cell is different. cur:";
         amap->template display_attribute<i>(a);
         std::cout<<" != first:";
-        amap->template display_attribute<i>(amap->template attribute<i>(it));
+        if (i==0 && reverseextremity)
+          amap->template display_attribute<i>(amap->template attribute<i>(amap->template beta<2>(it)));
+        else
+          amap->template display_attribute<i>(amap->template attribute<i>(it));
         std::cout<<" for dart ";
-        amap->display_dart(it);
+
+        if (i==0 && reverseextremity)
+          amap->display_dart(amap->template beta<2>(it));
+        else
+          amap->display_dart(it);
+
         std::cout<<std::endl;
         valid=false;
       }
 
-      if ( a!=amap->null_handle && it==amap->template dart_of_attribute<i>(a) )
+      if ( a!=amap->null_handle )
+        if ( (i==0 && reverseextremity && amap->template beta<2>(it)==amap->template dart_of_attribute<i>(a) ) ||
+             ((i>0 || !reverseextremity) && it==amap->template dart_of_attribute<i>(a) ) )
         found_dart=true;
 
       amap->mark(it, amark);
@@ -414,6 +440,38 @@ struct Decrease_attribute_functor
   { CGAL::internal::Decrease_attribute_functor_run<CMap,i>::run(amap, adart); }
 };
 // ****************************************************************************
+template<typename CMap, unsigned int i, typename T=
+         typename CMap::template Attribute_type<i>::type>
+struct Restricted_decrease_attribute_functor_run
+{
+  static void run(CMap* amap, typename CMap::Dart_handle adart)
+  {
+    if ( amap->template attribute<i>(adart)!=CMap::null_handle )
+    {
+      amap->template get_attribute<i>(amap->template attribute<i>(adart)).
+        dec_nb_refs();
+    }
+  }
+};
+/// Specialization for void attributes.
+template<typename CMap, unsigned int i>
+struct Restricted_decrease_attribute_functor_run<CMap, i, CGAL::Void>
+{
+  static void run(CMap*, typename CMap::Dart_handle)
+  {}
+};
+// ****************************************************************************
+/// Functor used to call restricted_decrease_attribute_ref_counting<i>
+/// on each i-cell attribute enabled
+template<typename CMap>
+struct Restricted_decrease_attribute_functor
+{
+  template <unsigned int i>
+  static void run(CMap* amap, typename CMap::Dart_handle adart)
+  { CGAL::internal::Restricted_decrease_attribute_functor_run<CMap,i>::
+        run(amap, adart); }
+};
+// ****************************************************************************
 /// Functor used to initialize all attributes to NULL.
 template<typename CMap>
 struct Init_attribute_functor
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex.h
index 01ad03d704f..d0d4c7ab25c 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex.h
@@ -338,11 +338,15 @@ namespace CGAL {
     /** Test if the lcc is valid.
      * A Linear_cell_complex is valid if it is a valid Combinatorial_map with
      * an attribute associated to each dart.
+     * @param reverseextremity to inverse the convention between source and
+     *        target of a dart. With false (default), a dart is associated with
+     *        a 0-attribute for its source (origin);
+     *        with true this is for its target (as in hds or surface mesh).
      * @return true iff the map is valid.
      */
-    bool is_valid() const
+    bool is_valid(bool reverseextremity=false) const
     {
-      bool valid = Base::is_valid();
+      bool valid = Base::is_valid(reverseextremity);
       for (typename Dart_range::const_iterator it(this->darts().begin()),
              itend(this->darts().end()); valid && it != itend; ++it)
       {
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex_incremental_builder_v2.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex_incremental_builder_v2.h
index f2b07afef1d..7d1f1457d9a 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex_incremental_builder_v2.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex_incremental_builder_v2.h
@@ -74,8 +74,8 @@ namespace CGAL {
 
         if (cur==lcc.null_handle)
         {
-          cur = lcc.create_dart(vertex_map[prev_vertex]);
-          Dart_handle opposite=lcc.create_dart(vertex_map[i]);
+          cur = lcc.create_dart(vertex_map[i]);
+          Dart_handle opposite=lcc.create_dart(vertex_map[prev_vertex]);
           lcc.template basic_link_beta_for_involution<2>(cur, opposite);
           add_dart_in_vertex_to_dart_map( opposite, i );
         }
@@ -126,7 +126,6 @@ namespace CGAL {
     // End of the surface. Return one dart of the created surface.
     Dart_handle end_surface()
     {
-      std::cout<<"******************";lcc.display_characteristics(std::cout)<<std::endl;
       unsigned int nb=0;
 
       for (typename LCC::Dart_range::iterator it=lcc.darts().begin(),
@@ -143,12 +142,13 @@ namespace CGAL {
           while (lcc.template attribute<2>(lcc.template beta<2>(other))!=NULL);
           assert(lcc.template is_free<0>(lcc.template beta<2>(other)));
           lcc.basic_link_beta_1(it, lcc.template beta<2>(other));
+
+          // For BGL halfedge graph, darts of border vertices must be border darts.
+          lcc.template set_dart_of_attribute<0>(lcc.vertex_attribute(it), it);
           ++nb;
         }
       }
 
-      std::cout<<"****************** nb sew="<<nb<<std::endl;
-      std::cout<<"******************";lcc.display_characteristics(std::cout)<<std::endl;
       return first_dart;
     }
 
@@ -166,7 +166,7 @@ namespace CGAL {
       Vertex_attribute_handle vh = vertex_map[j];
       for ( ; it!=itend; ++it )
       {
-        if ( lcc.temp_vertex_attribute(lcc.template beta<2>(*it))==vh )
+        if ( lcc.temp_vertex_attribute(*it)==vh )
           return (*it);
       }
       return lcc.null_handle;