diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map.h b/Combinatorial_map/include/CGAL/Combinatorial_map.h
index 967638fec29..ecaa92b0c0f 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map.h
@@ -1693,6 +1693,7 @@ namespace CGAL {
       int m=get_new_mark();
       std::deque<Dart_handle> dartv;
       std::deque<Dart_handle> modified_darts;
+      std::deque<Dart_handle> modified_darts2;
 
       for ( CMap_dart_iterator_basic_of_cell<Self,0> it(*this, adart, m);
            it.cont(); ++it)
@@ -1707,12 +1708,12 @@ namespace CGAL {
         if ( is_marked(it, m) )
         {
           modified_darts.push_back(it);
-          modified_darts.push_back(it->template beta<0>());
+          modified_darts2.push_back(it->template beta<0>());
           unlink_beta_0(it);
         }
         else
         {
-          modified_darts.push_back(it);
+          modified_darts2.push_back(it);
           modified_darts.push_back(it->template beta<1>());
           unlink_beta_1(it);
         }
@@ -1725,12 +1726,11 @@ namespace CGAL {
       CGAL_assertion( is_whole_map_unmarked(m) );
       free_mark(m);
 
-      // TODO pass the mark m to know which dart is modified by b0 of b1
       // We test the split of all the incident cells for all the non
       // void attributes.
       Helper::template Foreach_enabled_attributes_except
           <internal::Test_split_attribute_functor<Self,0>, 1>::
-          run(this, &modified_darts);
+          run(this, modified_darts, modified_darts2);
     }
 
     /** Unsew by beta1 the given dart plus all the required darts
@@ -1748,6 +1748,7 @@ namespace CGAL {
       int m = get_new_mark();
       std::deque<Dart_handle> dartv;
       std::deque<Dart_handle> modified_darts;
+      std::deque<Dart_handle> modified_darts2;
 
       for ( CMap_dart_iterator_basic_of_cell<Self,0> it(*this,adart,m);
            it.cont(); ++it)
@@ -1761,14 +1762,14 @@ namespace CGAL {
       {
         if ( is_marked(it, m) )
         {
-          modified_darts.push_back(it);
+          modified_darts2.push_back(it);
           modified_darts.push_back(it->template beta<1>());
           unlink_beta_1(it);
         }
         else
         {
           modified_darts.push_back(it);
-          modified_darts.push_back(it->template beta<0>());
+          modified_darts2.push_back(it->template beta<0>());
           unlink_beta_0(it);
         }
       }
@@ -1784,7 +1785,7 @@ namespace CGAL {
       // void attributes.
       Helper::template Foreach_enabled_attributes_except
           <internal::Test_split_attribute_functor<Self,1>, 1>::
-          run(this, &modified_darts);
+          run(this, modified_darts, modified_darts2);
     }
 
     /** Unsew by betai the given dart plus all the required darts
@@ -1816,7 +1817,7 @@ namespace CGAL {
       // void attributes.
       Helper::template Foreach_enabled_attributes_except
           <internal::Test_split_attribute_functor<Self,i>, i>::
-          run(this, &modified_darts);
+          run(this, modified_darts);
     }
 
     /** Unsew by betai the given dart plus all the required darts
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_insertions.h b/Combinatorial_map/include/CGAL/Combinatorial_map_insertions.h
index 2fb52811ac2..67e14648dfd 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_insertions.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_insertions.h
@@ -48,7 +48,7 @@ insert_cell_0_in_cell_1(CMap& amap, typename CMap::Dart_handle adart)
       }
     }
 
-    amap.template link_beta_1(*it, d1);
+    amap.link_beta_1(*it, d1);
     // TODO remove this group, and use link_beta instead ?
     //amap.group_all_dart_attributes_except(*it, d1, 1);
 
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h b/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h
index d61669525ed..05632564998 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h
@@ -104,13 +104,14 @@ namespace CGAL {
       // the operation.
       typename std::deque<typename CMap::Dart_handle>::iterator it =
           to_erase.begin();
+      //TODO remove ? or be sure that this is required
       for (; it != to_erase.end(); ++it)
         amap.update_dart_of_all_attributes(*it, mark);
 
       std::deque<typename CMap::Dart_handle> modified_darts;
 
       // For each dart of the i-cell, we modify i-links of neighbors.
-      for ( it=to_erase.begin(); it != to_erase.end(); ++it)
+      for ( it=to_erase.begin(); it!=to_erase.end(); ++it )
       {
         d1=(*it)->template beta<iinv>();
         while ( d1!=CMap::null_dart_handle && amap.is_marked(d1, mark) )
@@ -141,6 +142,7 @@ namespace CGAL {
                 amap.mark(d2, mark_modified_darts);
                 modified_darts.push_back(d1);
                 modified_darts.push_back(d2);
+                // TODO push only one out of two dart ?
                 /*if ( i==1 )
                 {
                   d2->basic_link_beta(d1, 0);
@@ -150,9 +152,9 @@ namespace CGAL {
               }
               else
               {
-                if ( !d1->is_free(i) )
+                if ( !d1->template is_free<i>() )
                 {
-                  d1->unlink_beta(i);
+                  d1->template unlink_beta<i>();
                   CGAL_assertion( !amap.is_marked(d1, mark_modified_darts) );
                   amap.mark(d1, mark_modified_darts);
                   modified_darts.push_back(d1);
@@ -161,9 +163,9 @@ namespace CGAL {
             }
             else if ( d2!=CMap::null_dart_handle )
             {
-              if ( !d2->is_free(iinv) )
+              if ( !d2->template is_free<iinv>() )
               {
-                d2->unlink_beta(iinv);
+                d2->template unlink_beta<iinv>();
                 CGAL_assertion( !amap.is_marked(d2, mark_modified_darts) );
                 amap.mark(d2, mark_modified_darts);
                 modified_darts.push_back(d2);
@@ -171,12 +173,13 @@ namespace CGAL {
             }
           }
         }
-        if ( (*it)->is_free(i+1) && !(*it)->is_free(i) )
+        if ( (*it)->template is_free<i+1>() &&
+             !(*it)->template is_free<i>() )
         {
-          d1 = (*it)->beta(i);
-          if ( !d1->is_free(iinv) )
+          d1 = (*it)->template beta<i>();
+          if ( !d1->template is_free<iinv>() )
           {
-            d1->unlink_beta(iinv);
+            d1->template unlink_beta<iinv>();
             CGAL_assertion( !amap.is_marked(d1, mark_modified_darts) );
             amap.mark(d1, mark_modified_darts);
             modified_darts.push_back(d1);
@@ -188,7 +191,7 @@ namespace CGAL {
       // void attributes.
       CMap::Helper::template Foreach_enabled_attributes_except
           <internal::Test_split_attribute_functor<CMap,i>, i>::
-                run(&amap, &modified_darts, mark_modified_darts);
+                run(&amap, modified_darts, mark_modified_darts);
 
       // We remove all the darts of the i-cell.
       for (  it=to_erase.begin(); it!=to_erase.end(); ++it )
@@ -233,18 +236,6 @@ namespace CGAL {
 
       std::deque<typename CMap::Dart_handle> modified_darts;
 
-/*      int mark_for_incident_cells[CMap::Helper::nb_attribs];
-      std::deque<std::deque<typename CMap::Dart_handle> >
-          incident_cells[CMap::Helper::nb_attribs];
-
-      // Marks used to mark all the incident cells.
-      for (int j=0; j<CMap::Helper::nb_attribs; ++j)
-      {
-        mark_for_incident_cells[j] = amap.get_new_mark();
-        CGAL_assertion( mark_for_incident_cells[j]!=-1 );
-      }
-*/
-
       // 1) We mark all the darts of the d-cell.
       for (CMap_dart_iterator_basic_of_cell<CMap,CMap::dimension>
            it(amap,adart,mark); it.cont(); ++it)
@@ -257,29 +248,13 @@ namespace CGAL {
       typename std::deque<typename CMap::Dart_handle>::iterator
         it = to_erase.begin();
 
-      // 2) We store all the incident cells that can be split by the operation,
-      // and we update the dart of the cells incident to the remove volume.
-/*      for ( ; it != to_erase.end(); ++it )
-      {
-        if ( !(*it)->is_free(CMap::dimension) )
-        {
-          CMap::Helper::template Foreach_enabled_attributes
-              <internal::Store_incident_cells<CMap,i> >::
-              run(&amap, *it, mark, &mark_for_incident_cells[0],
-                  &incident_cells[0]);
-        }
-        amap.update_dart_of_all_attributes(*it, mark);
-      }*/
-
-
       // 3) We unlink all the darts of the volume for beta-d.
       for ( it = to_erase.begin(); it != to_erase.end(); ++it )
       {
-        if ( !(*it)->is_free(CMap::dimension) &&
-             !amap.is_marked((*it)->beta(CMap::dimension), mark) )
+        if ( !(*it)->template is_free<CMap::dimension>() &&
+             !amap.is_marked((*it)->template beta<CMap::dimension>(), mark) )
         {
-          modified_darts.push_back((*it)->beta(CMap::dimension));
-          //(*it)->beta(CMap::dimension)->unlink_beta(CMap::dimension);
+          modified_darts.push_back((*it)->template beta<CMap::dimension>());
           amap.template unlink_beta_for_involution<CMap::dimension>(*it);
         }
       }
@@ -287,11 +262,8 @@ namespace CGAL {
       // 4) We test the split of all the incident cells for all the non
       // void attributes.
       CMap::Helper::template Foreach_enabled_attributes_except
-                <internal::Test_split_attribute_functor<CMap,i>,
-          CMap::dimension>::
-                run(&amap, &modified_darts, -1); //, modified_darts2);
-                    //&mark_for_incident_cells[0],
-                    //&incident_cells[0]);
+          <internal::Test_split_attribute_functor<CMap,i>,
+          CMap::dimension>::run(&amap, modified_darts);
 
       // 5) We remove all the darts of the d-cell.
       for ( it = to_erase.begin(); it != to_erase.end(); ++it )
@@ -300,14 +272,6 @@ namespace CGAL {
       CGAL_assertion( amap.is_whole_map_unmarked(mark) );
       amap.free_mark(mark);
 
-      // We free the marks.
-/*      for (int j=0; j<CMap::Helper::nb_attribs; ++j)
-      {
-        CGAL_assertion( amap.is_whole_map_marked
-                        (mark_for_incident_cells [j]) );
-        amap.free_mark( mark_for_incident_cells [j] );
-      }*/
-
       CGAL_expensive_postcondition( amap.is_valid() );
       assert( amap.is_valid() ); // TO REMOVE
 
@@ -338,26 +302,15 @@ namespace CGAL {
       std::deque<typename CMap::Dart_handle> to_erase;
 
       std::deque<typename CMap::Dart_handle> modified_darts;
-
-/*      int mark_for_incident_cells[CMap::Helper::nb_attribs];
-      std::deque<std::deque<typename CMap::Dart_handle> >
-          incident_cells[CMap::Helper::nb_attribs];
-
-      // Marks used to mark all the incident cells.
-      for (int j=0; j<CMap::Helper::nb_attribs; ++j)
-      {
-        mark_for_incident_cells [j] = amap.get_new_mark();
-        CGAL_assertion( mark_for_incident_cells [j]!=-1 );
-      }
-*/
+      std::deque<typename CMap::Dart_handle> modified_darts2;
 
       // First we store and mark all the darts of the 0-cell to remove.
       for ( CMap_dart_iterator_basic_of_cell<CMap,0> it(amap,adart,mark);
             it.cont(); ++it )
       {
         to_erase.push_back(it);
-        if ( !it->is_free(0) && dg1==NULL )
-        { dg1=it; dg2=it->beta(0); }
+        if ( !it->template is_free<0>() && dg1==NULL )
+        { dg1=it; dg2=it->template beta<0>(); }
         amap.mark(it, mark);
         ++res;
       }
@@ -366,39 +319,6 @@ namespace CGAL {
       // the operation.
       typename std::deque<typename CMap::Dart_handle>::iterator it =
         to_erase.begin();
-/*      for (; it != to_erase.end(); ++it)
-      {
-        if ( !(*it)->is_free(0) )
-        {
-          if ( !(*it)->is_free(1) && (*it)->beta(0)!=(*it) )
-          {
-            if ( dg1==NULL ) { dg1=(*it)->beta(0); dg2=(*it)->beta(1); }
-            CMap::Helper::template Foreach_enabled_attributes
-                <internal::Store_incident_cells<CMap,0> >::
-                run(&amap, *it, mark, &mark_for_incident_cells[0],
-                    &incident_cells[0]);
-          }
-          else
-          {
-            CMap::Helper::template Foreach_enabled_attributes
-                <internal::Store_incident_cells<CMap,0> >::
-                run(&amap, *it, mark, &mark_for_incident_cells[0],
-                    &incident_cells[0]);
-          }
-        }
-        else
-        {
-          if ( !(*it)->is_free(1) )
-          {
-            CMap::Helper::template Foreach_enabled_attributes
-                <internal::Store_incident_cells<CMap,0> >::
-                run(&amap, (*it)->beta(1), mark, &mark_for_incident_cells[0],
-                    &incident_cells[0]);
-          }
-        }
-      }
-*/
-
       for (; it != to_erase.end(); ++it)
         amap.update_dart_of_all_attributes(*it, mark);
 
@@ -410,18 +330,21 @@ namespace CGAL {
       // For each dart of the vertex, we modify 0 and 1-links of neighbors.
       for ( it=to_erase.begin(); it != to_erase.end(); ++it)
       {
-        if ( !(*it)->is_free(0) )
+        if ( !(*it)->template is_free<0>() )
         {
-          if ( !(*it)->is_free(1) && (*it)->beta(0)!=(*it) )
+          if ( !(*it)->template is_free<1>() &&
+               (*it)->template beta<0>()!=(*it) )
           {
-            amap.template basic_link_beta<1>((*it)->beta(0), (*it)->beta(1));
-            modified_darts.push_back((*it)->beta(0));
-            modified_darts.push_back((*it)->beta(1));
+            amap.basic_link_beta_1((*it)->template beta<0>(),
+                                   (*it)->template beta<1>());
+            modified_darts.push_back((*it)->template beta<0>());
+            modified_darts2.push_back((*it)->template beta<1>());
+            // TODO push only one out of two dart ?
           }
           else
           {
-            (*it)->beta(0)->unlink_beta(1);
-            modified_darts.push_back((*it)->beta(0));
+            (*it)->template beta<0>()->template unlink_beta<1>();
+            modified_darts.push_back((*it)->template beta<0>());
           }
 
           for ( unsigned int j=2; j<=CMap::dimension; ++j )
@@ -430,17 +353,18 @@ namespace CGAL {
             {
               // TODO push these darts in modified_darts ?
               // not sure this is required
-              amap.basic_link_beta((*it)->beta(0), (*it)->beta(j), j);
+              amap.basic_link_beta((*it)->template beta<0>(),
+                                   (*it)->beta(j), j);
             //((*it)->beta(0))->basic_link_beta((*it)->beta(j),j);
             }
           }
         }
         else
         {
-          if ( !(*it)->is_free(1) )
+          if ( !(*it)->template is_free<1>() )
           {
-            (*it)->beta(1)->unlink_beta(0);
-            modified_darts.push_back((*it)->beta(1));
+            (*it)->template beta<1>()->template unlink_beta<0>();
+            modified_darts2.push_back((*it)->template beta<1>());
           }
 
           for ( unsigned int j=2; j<=CMap::dimension; ++j )
@@ -459,11 +383,8 @@ namespace CGAL {
       // void attributes.
       CMap::Helper::template Foreach_enabled_attributes_except
           <internal::Test_split_attribute_functor<CMap,0>, 1>::
-               // <internal::Test_split_with_deque<CMap,0> >::
                 run(&amap,
-                    &modified_darts, -1); //mark_modified_darts);
-                    //&mark_for_incident_cells[0],
-                    //&incident_cells[0]);
+                    modified_darts, modified_darts2);
 
       // We remove all the darts of the i-cell.
       for (  it=to_erase.begin(); it!=to_erase.end(); ++it )
@@ -472,15 +393,6 @@ namespace CGAL {
       CGAL_assertion( amap.is_whole_map_unmarked(mark) );
       amap.free_mark(mark);
 
-      // We free the marks.
- /*     for (int j=0; j<CMap::Helper::nb_attribs; ++j)
-      {
-        CGAL_assertion( amap.is_whole_map_marked
-                        (mark_for_incident_cells [j]) );
-        amap.free_mark( mark_for_incident_cells [j] );
-      }
-      */
-
       CGAL_expensive_postcondition( amap.is_valid() );
       assert( amap.is_valid() ); // TO REMOVEE
 
@@ -555,8 +467,8 @@ namespace CGAL {
             it.cont(); ++it )
       {
         to_erase.push_back(it);
-        if ( !it->is_free(i-1) && dg1==NULL )
-        { dg1=it; dg2=it->beta(i-1); }
+        if ( !it->template is_free<i-1>() && dg1==NULL )
+        { dg1=it; dg2=it->template beta<i-1>(); }
         amap.mark(it, mark);
         ++res;
       }
@@ -577,20 +489,21 @@ namespace CGAL {
       // For each dart of the i-cell, we modify i-links of neighbors.
       for ( it=to_erase.begin(); it!=to_erase.end(); ++it )
       {
-        d1 = (*it)->beta(i);
+        d1 = (*it)->template beta<i>();
         while ( d1!=CMap::null_dart_handle && amap.is_marked(d1, mark) )
         {
-          d1 = d1->beta(imuinv)->beta(i);
-          if (d1 == (*it)->beta(i)) d1 = CMap::null_dart_handle;
+          d1 = d1->template beta<imuinv>()->template beta<i>();
+          if (d1 == (*it)->template beta<i>()) d1 = CMap::null_dart_handle;
         }
 
         if ( !amap.is_marked(d1, mark_modified_darts) )
         {
-          d2 = (*it)->beta(i-1)->beta(i);
+          d2 = (*it)->template beta<i-1>()->template beta<i>();
           while ( d2!=CMap::null_dart_handle && amap.is_marked(d2, mark) )
           {
-            d2 = d2->beta(i-1)->beta(i);
-            if ( d2==(*it)->beta(i-1)->beta(i) ) d2=CMap::null_dart_handle;
+            d2 = d2->template beta<i-1>()->template beta<i>();
+            if ( d2==(*it)->template beta<i-1>()->template beta<i>() )
+              d2=CMap::null_dart_handle;
           }
 
           if ( !amap.is_marked(d2, mark_modified_darts) )
@@ -599,15 +512,16 @@ namespace CGAL {
             {
               if (d2 != CMap::null_dart_handle && d1!=d2 )
               {
-                amap.template basic_link_beta<i>(d1, d2);
+                amap.template basic_link_beta_for_involution<i>(d1, d2);
                 amap.mark(d1, mark_modified_darts);
                 amap.mark(d2, mark_modified_darts);
                 modified_darts.push_back(d1);
                 modified_darts.push_back(d2);
+                // TODO push only one out of two dart ?
               }
               else
               {
-                if ( !d1->is_free(i) )
+                if ( !d1->template is_free<i>() )
                 {
                   d1->unlink_beta(i);
                   CGAL_assertion( !amap.is_marked(d1, mark_modified_darts) );
@@ -645,7 +559,7 @@ namespace CGAL {
       // void attributes.
       CMap::Helper::template Foreach_enabled_attributes_except
           <internal::Test_split_attribute_functor<CMap,i>, i>::
-          run(&amap, &modified_darts, mark_modified_darts);
+          run(&amap, modified_darts, mark_modified_darts);
 
       // We remove all the darts of the i-cell.
       for (  it=to_erase.begin(); it!=to_erase.end(); ++it )
@@ -697,6 +611,7 @@ namespace CGAL {
       std::deque<typename CMap::Dart_handle> to_erase;
 
       std::deque<typename CMap::Dart_handle> modified_darts;
+      std::deque<typename CMap::Dart_handle> modified_darts2;
 
       // First we store and mark all the darts of the 1-cell to contract.
       for ( CMap_dart_iterator_basic_of_cell<CMap,1> it(amap,adart,mark);
@@ -731,15 +646,16 @@ namespace CGAL {
             {
               /* modified_darts.push_back((*it)->beta(0));
               if ( (*it)->beta(0)!=(*it)->beta(1) )*/
-              modified_darts.push_back((*it)->beta(1));
-              amap.template basic_link_beta<1>((*it)->beta(0), (*it)->beta(1));
+              modified_darts2.push_back((*it)->template beta<1>());
+              amap.basic_link_beta_1((*it)->template beta<0>(),
+                                     (*it)->template beta<1>());
             }
           }
           else
           {
             // TODO todegroup.push(Dart_pair((*it)->beta(0), *it));
-            modified_darts.push_back((*it)->beta(0));
-            (*it)->beta(0)->unlink_beta(1);
+            modified_darts.push_back((*it)->template beta<0>());
+            (*it)->template beta<0>()->template unlink_beta<1>();
           }
         }
         else
@@ -747,8 +663,8 @@ namespace CGAL {
           if ( !(*it)->is_free(1) )
           {
             // TODO todegroup.push(Dart_pair((*it)->beta(1), *it));
-            modified_darts.push_back((*it)->beta(1));
-            (*it)->beta(1)->unlink_beta(0);
+            modified_darts2.push_back((*it)->template beta<1>());
+            (*it)->template beta<1>()->template unlink_beta<0>();
           }
         }
       }
@@ -757,11 +673,7 @@ namespace CGAL {
       // void attributes.
       CMap::Helper::template Foreach_enabled_attributes_except
           <internal::Test_split_attribute_functor<CMap,1>, 1>::
-               // <internal::Test_split_with_deque<CMap,0> >::
-                run(&amap,
-                    &modified_darts, -1); //mark_modified_darts);
-                    //&mark_for_incident_cells[0],
-                    //&incident_cells[0]);
+          run(&amap, modified_darts, modified_darts2);
 
       // 6) We remove all the darts of the cell.
       for (it = to_erase.begin(); it != to_erase.end(); ++it)
diff --git a/Combinatorial_map/include/CGAL/internal/Combinatorial_map_functors.h b/Combinatorial_map/include/CGAL/internal/Combinatorial_map_functors.h
index 4095411f046..ee1646e9bad 100644
--- a/Combinatorial_map/include/CGAL/internal/Combinatorial_map_functors.h
+++ b/Combinatorial_map/include/CGAL/internal/Combinatorial_map_functors.h
@@ -428,6 +428,16 @@ namespace CGAL
                     typename CMap::Dart_handle)
     {}
   };
+  // Specialization for i=1 and j=0. Do nothing as edges attributes are not
+  // modify when we modify beta_0.
+  template<typename CMap, typename T>
+  struct Group_attribute_functor_of_dart_run<CMap,1,0,T>
+  {
+    static void run(CMap*,
+                    typename CMap::Dart_handle,
+                    typename CMap::Dart_handle)
+    {}
+  };
   // **************************************************************************
   /// Functor used for link_beta to update the attributes of
   /// adart2 on the attributes of this dart, except for j-attributes.
@@ -660,6 +670,16 @@ namespace CGAL
                     typename CMap::Dart_handle)
     {}
   };
+  // Specialization for i=1 and j=0. Do nothing as edges attributes are not
+  // modify when we modify beta_0.
+  template<typename CMap, typename T>
+  struct Group_attribute_functor_run<CMap,1,0,T>
+  {
+    static void run(CMap*,
+                    typename CMap::Dart_handle,
+                    typename CMap::Dart_handle)
+    {}
+  };
   // **************************************************************************
   /// Functor used for sew to update the attributes of
   /// adart2 on the attributes of this dart, except for j-attributes.
@@ -804,85 +824,78 @@ namespace CGAL
     {}
   };
   // **************************************************************************
-  /// Functor used by Test_split_attribute_functor_run to process one dart.
-  template<typename CMap>
-  struct Test_split_attribute_functor_one_dart
+  // Function used by Test_split_attribute_functor_run to process one dart.
+  // Test the split of the i-cell containing the given dart adart.
+  // When we process a dart, we search in the Unique_hash_map if its
+  // i-attribute was already found. If yes, it means that we already
+  // found an i-cell with this attribute, thus this attribute is split.
+  // We mark (with mark) all the darts of the i-cell containing adart to
+  // process them exactly once.
+  template<typename CMap, unsigned int i>
+  void test_split_attribute_functor_one_dart
+  ( CMap* amap, typename CMap::Dart_handle adart,
+    Unique_hash_map<typename CMap::Helper::template Attribute_handle<i>::type,
+    unsigned int> & found_attributes, int mark )
   {
+    CGAL_assertion( amap!=NULL );
+    CGAL_assertion( adart!=NULL );
+    CGAL_static_assertion_msg(CMap::Helper::template
+                              Dimension_index<i>::value>=0,
+                              "Test_split_attribute_functor_one_dart<i> but "
+                              "i-attributes are disabled");
+
     typedef typename CMap::Dart_handle Dart_handle;
+    typedef typename CMap::Helper::template Attribute_handle<i>::type
+        Attribute_handle_i;
 
-    // Test the split of the i-cell containing the given dart adart.
-    // When we process a dart, we search in the Unique_hash_map if its
-    // i-attribute was already found. If yes, it means that we already
-    // found an i-cell with this attribute, thus this attribute is split.
-    // We mark (with mark) all the darts of the i-cell containing adart to
-    // process them exactly once.
-    template<unsigned int i>
-    static void run( CMap* amap,
-                     Dart_handle adart,
-                     Unique_hash_map<typename CMap::Helper::template
-                     Attribute_handle<i>::type, int> & found_attributes,
-                     int mark )
-    {
-      CGAL_assertion( amap!=NULL );
-      CGAL_assertion( adart!=NULL );
-      CGAL_static_assertion_msg(CMap::Helper::template
-                                Dimension_index<i>::value>=0,
-                                "Test_split_attribute_functor_one_dart<i> but "
-                                "i-attributes are disabled");
+    // If the current dart has no attribute, or if it is aldready marked,
+    // nothing to do.
+    if ( adart->template attribute<i>()==NULL ||
+         amap->is_marked(adart, mark) )
+      return;
 
-      typedef typename CMap::Helper::template Attribute_handle<i>::type
-      Attribute_handle_i;
+    Attribute_handle_i a1 = adart->template attribute<i>();
+    if ( found_attributes.is_defined(a1) )
+    {  // Here the attribute was already present in the hash_map
+      Attribute_handle_i a2 = amap->template create_attribute<i>(*a1);
+      a2->set_dart(adart);
 
-      // If the current dart has no attribute, or if it is aldready marked,
-      // nothing to do.
-      if ( adart->template attribute<i>()==NULL ||
-           amap->is_marked(adart, mark) )
-        return;
-
-      Attribute_handle_i a1 = adart->template attribute<i>();
-      if ( found_attributes.is_defined(a1) )
-      {  // Here the attribute was already present in the hash_map
-        Attribute_handle_i a2 = amap->template create_attribute<i>(*a1);
-        a2->set_dart(adart);
-
-        for ( CMap_dart_iterator_basic_of_cell<CMap, i>
-              itj(*amap, adart, mark); itj.cont(); ++itj )
-        {
-          amap->template set_attribute_of_dart<i>(itj, a2);
-          amap->mark(itj, mark);
-        }
-        Call_split_functor<CMap, i>::run(a1, a2);
-      }
-      else
+      for ( CMap_dart_iterator_basic_of_cell<CMap, i>
+            itj(*amap, adart, mark); itj.cont(); ++itj )
       {
-        // Here the attribute was not in the hash_map.
-        found_attributes[a1]=1;
-        a1->set_dart(adart);
+        amap->template set_attribute_of_dart<i>(itj, a2);
+        amap->mark(itj, mark);
+      }
+      Call_split_functor<CMap, i>::run(a1, a2);
+    }
+    else
+    {
+      // Here the attribute was not in the hash_map.
+      found_attributes[a1]=1;
+      a1->set_dart(adart);
 
-        for ( CMap_dart_iterator_basic_of_cell<CMap, i>
-              itj(*amap, adart, mark); itj.cont(); ++itj )
-        {
-          CGAL_assertion( itj->template attribute<i>()==a1 );
-          amap->mark(itj, mark);
-        }
+      for ( CMap_dart_iterator_basic_of_cell<CMap, i>
+            itj(*amap, adart, mark); itj.cont(); ++itj )
+      {
+        CGAL_assertion( itj->template attribute<i>()==a1 );
+        amap->mark(itj, mark);
       }
     }
-  };
+  }
   // **************************************************************************
   /// Functor used for unsew to test if i-attributes are split after an
   /// operation, except for j-attributes.
   ///   (j is the dimension of the beta modified for darts in modified_darts,
-  ///    if j==0 modified_darts_2 are the darts modified for beta_1).
+  ///    if j==0 modified_darts2 are the darts modified for beta_1).
   template<typename CMap, unsigned int i, unsigned int j=CMap::dimension+1,
            typename T=
            typename CMap::Helper::template Attribute_type<i>::type>
   struct Test_split_attribute_functor_run
   {
+    // modified_darts is the set of modified darts for beta_j
     static void run( CMap* amap,
-                     std::deque<typename CMap::Dart_handle>
-                     *modified_darts,
-                     std::deque<typename CMap::Dart_handle>
-                     */*modified_darts2*/,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts,
                      int mark_modified_darts=-1)
     {
       CGAL_static_assertion( 1<=i && i<=CMap::dimension );
@@ -899,23 +912,86 @@ namespace CGAL
       Unique_hash_map<Attribute_handle_i, unsigned int> found_attributes;
 
       int mark = amap->get_new_mark(); // to mark incident cells.
-      typename std::deque<typename CMap::Dart_handle>::iterator
-          it=modified_darts->begin();
-      for ( ; it!=modified_darts->end(); ++it )
+      typename std::deque<typename CMap::Dart_handle>::const_iterator
+          it=modified_darts.begin();
+      for ( ; it!=modified_darts.end(); ++it )
       {
-        Test_split_attribute_functor_one_dart<CMap>::
-            run<i>(amap, *it, found_attributes, mark);
+        test_split_attribute_functor_one_dart<CMap,i>
+            (amap, *it, found_attributes, mark);
       }
 
       // Now we unmark all the marked darts.
       amap->negate_mark(mark);
-      for ( it=modified_darts->begin(); it!=modified_darts->end(); ++it )
+      for ( it=modified_darts.begin(); it!=modified_darts.end(); ++it )
       {
         if ( mark_modified_darts!=-1 )
           amap->unmark(*it, mark_modified_darts);
 
         if ( !amap->is_marked(*it, mark) )
-          mark_cell<i>(*amap, *it, mark);
+          mark_cell<CMap, i>(*amap, *it, mark);
+      }
+
+      CGAL_assertion( amap->is_whole_map_marked(mark) );
+      amap->free_mark(mark);
+    }
+    static void run( CMap* amap,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts2,
+                     int mark_modified_darts=-1)
+    {
+      // Normalement on utilise la version avec modified_darts2
+      // que dans le cas j=0
+      CGAL_assertion(false);
+      // TODO remove cette function run
+
+      CGAL_static_assertion( 1<=i && i<=CMap::dimension );
+      CGAL_assertion( i!=j );
+      CGAL_assertion( amap!=NULL );
+      CGAL_static_assertion_msg(CMap::Helper::template
+                                Dimension_index<i>::value>=0,
+                                "Test_split_attribute_functor_run<i> but "
+                                "i-attributes are disabled");
+
+      typedef typename CMap::Helper::template Attribute_handle<i>::type
+      Attribute_handle_i;
+
+      Unique_hash_map<Attribute_handle_i, unsigned int> found_attributes;
+
+      int mark = amap->get_new_mark(); // to mark incident cells.
+      typename std::deque<typename CMap::Dart_handle>::const_iterator
+          it=modified_darts.begin();
+      for ( ; it!=modified_darts.end(); ++it )
+      {
+        test_split_attribute_functor_one_dart<CMap,i>
+            (amap, *it, found_attributes, mark);
+      }
+      typename std::deque<typename CMap::Dart_handle>::const_iterator
+          it2=modified_darts2.begin();
+      for ( ; it2!=modified_darts2.end(); ++it2 )
+      {
+        test_split_attribute_functor_one_dart<CMap,i>
+            (amap, *it2, found_attributes, mark);
+      }
+
+      // Now we unmark all the marked darts.
+      amap->negate_mark(mark);
+      for ( it=modified_darts.begin(); it!=modified_darts.end(); ++it )
+      {
+        if ( mark_modified_darts!=-1 )
+          amap->unmark(*it, mark_modified_darts);
+
+        if ( !amap->is_marked(*it, mark) )
+          mark_cell<CMap, i>(*amap, *it, mark);
+      }
+      for ( it2=modified_darts2.begin(); it2!=modified_darts2.end(); ++it2 )
+      {
+        if ( mark_modified_darts!=-1 )
+          amap->unmark(*it2, mark_modified_darts);
+
+        if ( !amap->is_marked(*it2, mark) )
+          mark_cell<CMap, i>(*amap, *it2, mark);
       }
 
       CGAL_assertion( amap->is_whole_map_marked(mark) );
@@ -927,10 +1003,8 @@ namespace CGAL
   struct Test_split_attribute_functor_run<CMap, 0, j, T>
   {
     static void run( CMap* amap,
-                     std::deque<typename CMap::Dart_handle>
-                     *modified_darts,
-                     std::deque<typename CMap::Dart_handle>
-                     */*modified_darts2*/,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts,
                      int mark_modified_darts=-1)
     {
       CGAL_assertion( j!=0 && j!=1 );
@@ -947,32 +1021,113 @@ namespace CGAL
       typename CMap::Dart_handle od=NULL;
 
       int mark = amap->get_new_mark(); // to mark incident cells.
-      typename std::deque<typename CMap::Dart_handle>::iterator
-          it=modified_darts->begin();
-      for ( ; it!=modified_darts->end(); ++it )
+      typename std::deque<typename CMap::Dart_handle>::const_iterator
+          it=modified_darts.begin();
+      for ( ; it!=modified_darts.end(); ++it )
       {
-        Test_split_attribute_functor_one_dart<CMap>::
-            run<0>(amap, *it, found_attributes, mark);
+        test_split_attribute_functor_one_dart<CMap,0>
+            (amap, *it, found_attributes, mark);
 
         od=(*it)->other_extremity();
         if ( od!=NULL )
-          Test_split_attribute_functor_one_dart<CMap>::
-              run<0>(amap, od, found_attributes, mark);
+          test_split_attribute_functor_one_dart<CMap,0>
+              (amap, od, found_attributes, mark);
       }
 
       // Now we unmark all the marked darts.
       amap->negate_mark(mark);
-      for ( it=modified_darts->begin(); it!=modified_darts->end(); ++it )
+      for ( it=modified_darts.begin(); it!=modified_darts.end(); ++it )
       {
         if ( mark_modified_darts!=-1 )
           amap->unmark(*it, mark_modified_darts);
 
         if ( !amap->is_marked(*it, mark) )
-          mark_cell<0>(*amap, *it, mark);
+          mark_cell<CMap, 0>(*amap, *it, mark);
 
         od=(*it)->other_extremity();
         if ( od!=NULL && !amap->is_marked(od, mark) )
-          mark_cell<0>(*amap, od, mark);
+          mark_cell<CMap, 0>(*amap, od, mark);
+      }
+
+      CGAL_assertion( amap->is_whole_map_marked(mark) );
+      amap->free_mark(mark);
+    }
+    static void run( CMap* amap,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts2,
+                     int mark_modified_darts=-1)
+    {
+      // Normalement on utilise la version avec modified_darts2
+      // que dans le cas j=0
+      CGAL_assertion(false);
+      // TODO remove cette function run
+
+      CGAL_assertion( j!=0 && j!=1 );
+      CGAL_assertion( amap!=NULL );
+      CGAL_static_assertion_msg(CMap::Helper::template
+                                Dimension_index<0>::value>=0,
+                                "Test_split_attribute_functor_run<0> but "
+                                "0-attributes are disabled");
+
+      typedef typename CMap::Helper::template Attribute_handle<0>::type
+          Attribute_handle_0;
+
+      Unique_hash_map<Attribute_handle_0, unsigned int> found_attributes;
+      typename CMap::Dart_handle od=NULL;
+
+      int mark = amap->get_new_mark(); // to mark incident cells.
+      typename std::deque<typename CMap::Dart_handle>::const_iterator
+          it=modified_darts.begin();
+      for ( ; it!=modified_darts.end(); ++it )
+      {
+        test_split_attribute_functor_one_dart<CMap,0>
+            (amap, *it, found_attributes, mark);
+
+        od=(*it)->other_extremity();
+        if ( od!=NULL )
+          test_split_attribute_functor_one_dart<CMap,0>
+              (amap, od, found_attributes, mark);
+      }
+      typename std::deque<typename CMap::Dart_handle>::const_iterator
+          it2=modified_darts2.begin();
+      for ( ; it2!=modified_darts2.end(); ++it2 )
+      {
+        test_split_attribute_functor_one_dart<CMap,0>
+            (amap, *it2, found_attributes, mark);
+
+        od=(*it2)->other_extremity();
+        if ( od!=NULL )
+          test_split_attribute_functor_one_dart<CMap,0>
+              (amap, od, found_attributes, mark);
+      }
+
+      // Now we unmark all the marked darts.
+      amap->negate_mark(mark);
+      for ( it=modified_darts.begin(); it!=modified_darts.end(); ++it )
+      {
+        if ( mark_modified_darts!=-1 )
+          amap->unmark(*it, mark_modified_darts);
+
+        if ( !amap->is_marked(*it, mark) )
+          mark_cell<CMap, 0>(*amap, *it, mark);
+
+        od=(*it)->other_extremity();
+        if ( od!=NULL && !amap->is_marked(od, mark) )
+          mark_cell<CMap, 0>(*amap, od, mark);
+      }
+      for ( it2=modified_darts2.begin(); it2!=modified_darts2.end(); ++it2 )
+      {
+        if ( mark_modified_darts!=-1 )
+          amap->unmark(*it2, mark_modified_darts);
+
+        if ( !amap->is_marked(*it2, mark) )
+          mark_cell<CMap, 0>(*amap, *it2, mark);
+
+        od=(*it2)->other_extremity();
+        if ( od!=NULL && !amap->is_marked(od, mark) )
+          mark_cell<CMap, 0>(*amap, od, mark);
       }
 
       CGAL_assertion( amap->is_whole_map_marked(mark) );
@@ -983,18 +1138,115 @@ namespace CGAL
   template<typename CMap, typename T>
   struct Test_split_attribute_functor_run<CMap, 0, 0, T>
   {
+    static void run( CMap* amap,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts2,
+                     int mark_modified_darts=-1)
+    {
+      CGAL_assertion( amap!=NULL );
+      CGAL_static_assertion_msg(CMap::Helper::template
+                                Dimension_index<0>::value>=0,
+                                "Test_split_attribute_functor_run<0> but "
+                                "0-attributes are disabled");
+
+      typedef typename CMap::Helper::template Attribute_handle<0>::type
+          Attribute_handle_0;
+
+      Unique_hash_map<Attribute_handle_0, unsigned int> found_attributes;
+      typename CMap::Dart_handle od=NULL;
+
+      int mark = amap->get_new_mark(); // to mark incident cells.
+      typename std::deque<typename CMap::Dart_handle>::const_iterator
+          it=modified_darts.begin();
+      for ( ; it!=modified_darts.end(); ++it )
+      {
+        test_split_attribute_functor_one_dart<CMap,0>
+            (amap, *it, found_attributes, mark);
+      }
+      typename std::deque<typename CMap::Dart_handle>::const_iterator
+          it2=modified_darts2.begin();
+      for ( ; it2!=modified_darts2.end(); ++it2 )
+      {
+        od=(*it2)->other_extremity();
+        if ( od!=NULL )
+          test_split_attribute_functor_one_dart<CMap,0>
+              (amap, od, found_attributes, mark);
+      }
+
+      // Now we unmark all the marked darts.
+      amap->negate_mark(mark);
+      for ( it=modified_darts.begin(); it!=modified_darts.end(); ++it )
+      {
+        if ( mark_modified_darts!=-1 )
+          amap->unmark(*it, mark_modified_darts);
+
+        if ( !amap->is_marked(*it, mark) )
+          mark_cell<CMap, 0>(*amap, *it, mark);
+      }
+      for ( it2=modified_darts2.begin(); it2!=modified_darts2.end(); ++it2 )
+      {
+        if ( mark_modified_darts!=-1 )
+          amap->unmark(*it2, mark_modified_darts);
+
+        od=(*it2)->other_extremity();
+        if ( od!=NULL && !amap->is_marked(od, mark) )
+          mark_cell<CMap, 0>(*amap, od, mark);
+      }
+
+      CGAL_assertion( amap->is_whole_map_marked(mark) );
+      amap->free_mark(mark);
+    }
   };
-  // Specialization for i=1 and j=0.
+  // Specialization for i=0 and j=1.
+  // Equivalent to i=0 and j=0.
   template<typename CMap, typename T>
-  struct Test_split_attribute_functor_run<CMap, 1, 0, T>
+  struct Test_split_attribute_functor_run<CMap, 0, 1, T>
   {
-    // No sense as we use always <0,0> ?
+    static void run( CMap*, const std::deque<typename CMap::Dart_handle>&,
+                     int=-1)
+    { CGAL_assertion(false); }
+    static void run( CMap*, const std::deque<typename CMap::Dart_handle>&,
+                     const std::deque<typename CMap::Dart_handle>&, int=-1)
+    { CGAL_assertion(false); }
   };
   // Specialization for void attributes.
   template<typename CMap, unsigned int i, unsigned int j>
   struct Test_split_attribute_functor_run<CMap, i, j, CGAL::Void>
   {
+    static void run( CMap*, const std::deque<typename CMap::Dart_handle>&,
+                     int=-1)
+    {}
+    static void run( CMap*, const std::deque<typename CMap::Dart_handle>&,
+                     const std::deque<typename CMap::Dart_handle>&, int=-1)
+    {}
   };
+  // Specialization for i=j.
+  template<typename CMap, unsigned int i, typename T>
+  struct Test_split_attribute_functor_run<CMap, i, i, T>
+  {
+    static void run( CMap*, const std::deque<typename CMap::Dart_handle>&,
+                     int=-1)
+    {}
+    static void run( CMap*, const std::deque<typename CMap::Dart_handle>&,
+                     const std::deque<typename CMap::Dart_handle>&, int=-1)
+    {}
+  };
+  // Specialization for i=1 and j=0 (edge attributes are not modified
+  // when we modify beta_0).
+  template<typename CMap, typename T>
+  struct Test_split_attribute_functor_run<CMap, 1, 0, T>
+  {
+    static void run( CMap*, const std::deque<typename CMap::Dart_handle>&,
+                     int=-1)
+    {}
+    static void run( CMap*, const std::deque<typename CMap::Dart_handle>&,
+                     const std::deque<typename CMap::Dart_handle>&, int=-1)
+    {}
+  };
+  // **************************************************************************
+
   /// Functor used for unsew to test if i-attributes are split after an
   /// operation, except for j-attributes.
   /// We define run<i> to allows to use this functor with
@@ -1002,116 +1254,33 @@ namespace CGAL
   template<typename CMap, unsigned int j=CMap::dimension+1>
   struct Test_split_attribute_functor
   {
-    typedef typename CMap::Dart_handle Dart_handle;
-
-    // Test the split of the i-cell containing dart adart.
-    template<unsigned int i>
-    static void test_one_dart( CMap* amap,
-                               Dart_handle adart,
-                               Unique_hash_map<typename CMap::Helper::template
-                               Attribute_handle<i>::type, int> &
-                               found_attributes, int mark )
-    {
-      typedef typename CMap::Helper::template Attribute_handle<i>::type
-      Attribute_handle_i;
-
-      if ( adart->template attribute<i>()!=NULL &&
-           !amap->is_marked(adart, mark) )
-      {
-        Attribute_handle_i a1 = adart->template attribute<i>();
-        if ( found_attributes.is_defined(a1) )
-        {  // Here the attribute was already present in the hash_map
-          Attribute_handle_i a2 = amap->template create_attribute<i>(*a1);
-
-          for ( CMap_dart_iterator_basic_of_cell<CMap, i>
-                itj(*amap, adart, mark); itj.cont(); ++itj )
-          {
-            amap->template set_attribute_of_dart<i>(itj, a2);
-            amap->mark(itj, mark);
-          }
-          a2->set_dart(adart);
-          Call_split_functor<CMap, i>::run(a1, a2);
-        }
-        else
-        {
-          // Here the attribute was not in the hash_map.
-          found_attributes[a1]=1;
-          a1->set_dart(adart);
-
-          for ( CMap_dart_iterator_basic_of_cell<CMap, i>
-                itj(*amap, adart, mark); itj.cont(); ++itj )
-          {
-            CGAL_assertion( itj->template attribute<i>()==a1 );
-            amap->mark(itj, mark);
-          }
-        }
-      }
-    }
-
     // Test the split of i-attributes, for all modified darts given in
     // modified_darts, and marked with mark_modified_darts.
     // For each split attribute, create a new i-attribute, associate
     // it with the new i-cell and call onsplit functors.
     template <unsigned int i>
     static void run( CMap* amap,
-                     std::deque<typename CMap::Dart_handle>
-                     *modified_darts,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts,
                      int mark_modified_darts=-1)
     {
-      CGAL_assertion( i!=j );
-      CGAL_assertion( amap!=NULL );
-
-      typedef typename CMap::Helper::template Attribute_handle<i>::type
-      Attribute_handle_i;
-
-      Unique_hash_map<Attribute_handle_i, unsigned int> found_attributes;
-
-      int mark = amap->get_new_mark(); // to mark incident cells.
-      for ( typename std::deque<Dart_handle>::iterator
-            it=modified_darts->begin(); it!=modified_darts->end(); ++it )
-      {
-        test_one_dart<i>(amap, *it, found_attributes, mark);
-
-        if ( /*j!=1 &&*/ i==0 ) // TODO Here we test too many cases, but more works to do
-        {
-          typename CMap::Dart_handle od = (*it)->other_extremity();
-          if ( od!=NULL )
-            test_one_dart<i>(amap, od, found_attributes, mark);
-        }
-      }
-
-      // Now we unmark all the marked darts.
-      amap->negate_mark(mark);
-      for ( typename std::deque<Dart_handle>::iterator
-            it=modified_darts->begin(); it!=modified_darts->end(); ++it )
-      {
-        if ( mark_modified_darts!=-1 )
-          amap->unmark(*it, mark_modified_darts);
-
-        if ( !amap->is_marked(*it, mark) )
-          for ( CMap_dart_iterator_basic_of_cell<CMap,i>
-                itj(*amap, *it, mark); itj.cont(); ++itj )
-          {
-            amap->mark(itj, mark);
-          }
-
-        if ( /*j!=1 &&*/ i==0 )
-        {
-          Dart_handle od = (*it)->other_extremity();
-          if ( od!=NULL && !amap->is_marked(od, mark) )
-            for ( CMap_dart_iterator_basic_of_cell<CMap,i>
-                  itj(*amap, od, mark); itj.cont(); ++itj )
-            {
-              amap->mark(itj, mark);
-            }
-        }
-      }
-
-      CGAL_assertion( amap->is_whole_map_marked(mark) );
-      amap->free_mark(mark);
+      Test_split_attribute_functor_run<CMap, i, j>::run(amap, modified_darts,
+                                                        mark_modified_darts);
+    }
+    template <unsigned int i>
+    static void run( CMap* amap,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts,
+                     const std::deque<typename CMap::Dart_handle>
+                     &modified_darts2,
+                     int mark_modified_darts=-1)
+    {
+      Test_split_attribute_functor_run<CMap, i, j>::run(amap, modified_darts,
+                                                        modified_darts2,
+                                                        mark_modified_darts);
     }
   };
-
+  // **************************************************************************
   } // namespace internal
 } // namespace CGAL