diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h b/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h
index 5b213ab0dd3..5f19046b29c 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h
@@ -220,7 +220,7 @@ namespace CGAL {
       CGAL_assertion( (is_removable<Map,i>(amap, adart)) );
 
       typename Map::Dart_handle d1, d2;
-      int mark  = amap.get_new_mark();
+      int mark = amap.get_new_mark();
       std::deque<typename Map::Dart_handle> to_erase;
 
       const int iinv = CGAL_BETAINV(i);
@@ -229,12 +229,14 @@ namespace CGAL {
       std::deque<std::deque<typename Map::Dart_handle> >
           incident_cells[Map::Helper::nb_attribs];
 
+      // Marks used to mark all the incident cells.
       for (int j=0; j<Map::Helper::nb_attribs; ++j)
       {
         mark_for_incident_cells [j] = amap.get_new_mark();
         CGAL_assertion( mark_for_incident_cells [j]!=-1 );
       }
 
+      // First we store and mark all the darts of the i-cell to remove.
       size_t res = 0;
       for ( CMap_dart_iterator_basic_of_cell<Map,i> it(amap,adart,mark);
             it.cont(); ++it )
@@ -246,6 +248,8 @@ namespace CGAL {
 
       typename Map::Dart_handle dg1=NULL, dg2=NULL;
 
+      // Second we store all the incident cells that can be split by
+      // the operation.
       typename std::deque<typename Map::Dart_handle>::iterator it =
         to_erase.begin();
       for (; it != to_erase.end(); ++it)
@@ -264,8 +268,6 @@ namespace CGAL {
           if ( d2==(*it)->beta(i+1)->beta(i) ) d2=Map::null_dart_handle;
         }
 
-        // TODO ? We can optimize by using map.basic_link_beta but we
-        // need to mark the second dart to not process another time...
         if (d1 != Map::null_dart_handle)
         {
           if (d2 != Map::null_dart_handle)
@@ -309,11 +311,11 @@ namespace CGAL {
         amap.update_dart_of_all_attributes(*it, mark);
       }
 
-      // 2) We group the two (i+1)-cells if they exist.
+      // We group the two (i+1)-cells incident if they exist.
       if ( dg1!=NULL )
         amap.template group_attribute<i+1>(dg1, dg2);
 
-      // 4) For each dart of the cell, we modify i-link of neighbors.
+      // 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_inv(i);
@@ -361,18 +363,21 @@ namespace CGAL {
         }
       }
 
-      // 6) We remove all the darts of the cell.
+      // We remove all the darts of the i-cell.
       for (  it=to_erase.begin(); it!=to_erase.end(); ++it )
       { amap.erase_dart(*it); }
 
       CGAL_assertion( amap.is_whole_map_unmarked(mark) );
       amap.free_mark(mark);
 
+      // We test the split of all the incident cells for all the non
+      // void attributes.
       Map::Helper::template Foreach_enabled_attributes
                 <internal::Test_split_with_deque<Map,i> >::
                 run(&amap, &mark_for_incident_cells[0],
                     &incident_cells[0]);
 
+      // We free the marks.
       for (int j=0; j<Map::Helper::nb_attribs; ++j)
       {
         CGAL_assertion( amap.is_whole_map_marked
@@ -380,7 +385,7 @@ namespace CGAL {
         amap.free_mark( mark_for_incident_cells [j] );
       }
 
-      // CGAL_postcondition(amap.is_valid());
+      CGAL_expensive_postcondition( amap.is_valid() );
 
       return res;
     }
@@ -398,14 +403,25 @@ namespace CGAL {
     {
       CGAL_assertion( adart!=NULL );
 
-      std::vector<typename Map::Dart_handle> to_erase;
+      std::deque<typename Map::Dart_handle> to_erase;
       int mark  = amap.get_new_mark();
       size_t res = 0;
 
       // Stack of couple of dart for which we must call degroup_all_attributes
       typedef std::pair<typename Map::Dart_handle, typename Map::Dart_handle>
         Dart_pair;
-      std::stack<Dart_pair> todegroup;
+      std::deque<Dart_pair> todegroup;
+
+      int mark_for_incident_cells[Map::Helper::nb_attribs];
+      std::deque<std::deque<typename Map::Dart_handle> >
+          incident_cells[Map::Helper::nb_attribs];
+
+      // Marks used to mark all the incident cells.
+      for (int j=0; j<Map::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.
       {
@@ -418,38 +434,55 @@ namespace CGAL {
         }
       }
 
-      // 2) We update the cells incident to the remove volume.
-      typename std::vector<typename Map::Dart_handle>::iterator
+      typename std::deque<typename Map::Dart_handle>::iterator
         it = to_erase.begin();
-      for (; it != to_erase.end(); ++it)
-      { amap.update_dart_of_all_attributes(*it, mark); }
+
+      // 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(Map::dimension) )
+        {
+          Map::Helper::template Foreach_enabled_attributes
+              <internal::Store_incident_cells<Map,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(Map::dimension) )
         {
-          todegroup.push(Dart_pair(*it, (*it)->beta(Map::dimension)));
           amap.unlink_beta_for_involution(*it,Map::dimension);
         }
       }
 
-      // 4) We degroup all the pairs
-      while ( !todegroup.empty() )
-      {
-        Dart_pair & p=todegroup.top();
-        amap.degroup_all_attributes(p.first,p.second);
-        todegroup.pop();
-      }
-
-      // 5) last, we remove all the darts of the d-cell.
+      // 4) We remove all the darts of the d-cell.
       for ( it = to_erase.begin(); it != to_erase.end(); ++it )
       { amap.erase_dart(*it); }
 
       CGAL_assertion( amap.is_whole_map_unmarked(mark) );
       amap.free_mark(mark);
 
-      //CGAL_postcondition(amap.is_valid());
+      // 5) We test the split of all the incident cells for all the non
+      // void attributes.
+      Map::Helper::template Foreach_enabled_attributes
+                <internal::Test_split_with_deque<Map,i> >::
+                run(&amap, &mark_for_incident_cells[0],
+                    &incident_cells[0]);
+
+      // We free the marks.
+      for (int j=0; j<Map::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() );
 
       return res;
     }
@@ -469,39 +502,87 @@ namespace CGAL {
 
       size_t res = 0;
 
-      // Stack of couple of dart for which we must call degroup_all_attributes
-      typedef std::pair<typename Map::Dart_handle, typename Map::Dart_handle>
-        Dart_pair;
-      std::stack<Dart_pair> todegroup;
-
-      // 1) We group the two edges if they exist.
-      if (!adart->is_free(0))
-        amap.template group_attribute<1>(adart, adart->beta(0));
-
       typename Map::Dart_handle d1, d2;
-      int mark    = amap.get_new_mark();
-      std::vector<typename Map::Dart_handle> to_erase;
+      int mark = amap.get_new_mark();
+      std::deque<typename Map::Dart_handle> to_erase;
 
-      // 2) We mark all the darts of the vertex.
+      int mark_for_incident_cells[Map::Helper::nb_attribs];
+      std::deque<std::deque<typename Map::Dart_handle> >
+          incident_cells[Map::Helper::nb_attribs];
+
+      // Marks used to mark all the incident cells.
+      for (int j=0; j<Map::Helper::nb_attribs; ++j)
       {
-        for ( CMap_dart_iterator_basic_of_cell<Map,0> it(amap,adart,mark);
-              it.cont(); ++it )
-        {
-          to_erase.push_back(it);
-          amap.mark(it,mark);
-          ++res;
-        }
+        mark_for_incident_cells [j] = amap.get_new_mark();
+        CGAL_assertion( mark_for_incident_cells [j]!=-1 );
       }
 
-      // 3) We modify the darts of the cells incident to the vertex
-      //    when they are marked to remove.
-      typename std::vector<typename Map::Dart_handle>::iterator
-        it = to_erase.begin();
-      for (; it != to_erase.end(); ++it)
-      { amap.update_dart_of_all_attributes(*it, mark); }
+      // First we store and mark all the darts of the 0-cell to remove.
+      for ( CMap_dart_iterator_basic_of_cell<Map,0> it(amap,adart,mark);
+            it.cont(); ++it )
+      {
+        to_erase.push_back(it);
+        amap.mark(it, mark);
+        ++res;
+      }
 
-      // 4) For each dart of the cell, we modify link of neighbors.
-      for ( it=to_erase.begin(); it!=to_erase.end(); ++it )
+      typename Map::Dart_handle dg1=NULL, dg2=NULL;
+
+      // Second we store all the incident cells that can be split by
+      // the operation.
+      typename std::deque<typename Map::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) )
+            amap.template basic_link_beta<1>((*it)->beta(0), (*it)->beta(1));
+          else
+          {
+            Map::Helper::template Foreach_enabled_attributes
+                <internal::Store_incident_cells<Map,0> >::
+                run(&amap, *it, mark, &mark_for_incident_cells[0],
+                    &incident_cells[0]);
+            // todegroup.push(Dart_pair((*it)->beta(0), *it));
+            // (*it)->beta(0)->unlink_beta(1);
+          }
+
+          /* for ( unsigned int j=2; j<=Map::dimension; ++j )
+          {
+            if ( !(*it)->is_free(j) )
+              amap.basic_link_beta((*it)->beta(0), (*it)->beta(j), j);
+            //((*it)->beta(0))->basic_link_beta((*it)->beta(j),j);
+          } */
+        }
+        else
+        {
+          if ( !(*it)->is_free(1) )
+          {
+            Map::Helper::template Foreach_enabled_attributes
+                <internal::Store_incident_cells<Map,0> >::
+                run(&amap, (*it)->beta(1), mark, &mark_for_incident_cells[0],
+                    &incident_cells[0]);
+            //todegroup.push(Dart_pair((*it)->beta(1), *it));
+            //(*it)->beta(1)->unlink_beta(0);
+          }
+
+          /* for ( unsigned int j=2; j<=Map::dimension; ++j )
+          {
+            if ( !(*it)->is_free(j) )
+              amap.unlink_beta(*it, j);
+          } */
+        }
+      }
+
+      // We group the two edges incident if they exist.
+      if ( dg1!=NULL )
+        amap.template group_attribute<1>(dg1, dg2);
+      //if (!adart->is_free(0))
+      //  amap.template group_attribute<1>(adart, adart->beta(0));
+
+      // 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) )
         {
@@ -509,7 +590,6 @@ namespace CGAL {
             amap.template basic_link_beta<1>((*it)->beta(0), (*it)->beta(1));
           else
           {
-            todegroup.push(Dart_pair((*it)->beta(0), *it));
             (*it)->beta(0)->unlink_beta(1);
           }
 
@@ -524,7 +604,6 @@ namespace CGAL {
         {
           if ( !(*it)->is_free(1) )
           {
-            todegroup.push(Dart_pair((*it)->beta(1), *it));
             (*it)->beta(1)->unlink_beta(0);
           }
 
@@ -536,22 +615,29 @@ namespace CGAL {
         }
       }
 
-      // 5) We degroup all the pairs
-      while ( !todegroup.empty() )
-      {
-        Dart_pair & p=todegroup.top();
-        amap.degroup_all_attributes(p.first,p.second);
-        todegroup.pop();
-      }
-
-      // 6) We remove all the darts of the cell.
-      for (it = to_erase.begin(); it != to_erase.end(); ++it)
+      // We remove all the darts of the i-cell.
+      for (  it=to_erase.begin(); it!=to_erase.end(); ++it )
       { amap.erase_dart(*it); }
 
       CGAL_assertion( amap.is_whole_map_unmarked(mark) );
       amap.free_mark(mark);
 
-      // CGAL_postcondition( amap.is_valid() );
+      // We test the split of all the incident cells for all the non
+      // void attributes.
+      Map::Helper::template Foreach_enabled_attributes
+                <internal::Test_split_with_deque<Map,0> >::
+                run(&amap, &mark_for_incident_cells[0],
+                    &incident_cells[0]);
+
+      // We free the marks.
+      for (int j=0; j<Map::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() );
 
       return res;
     }