diff --git a/Combinatorial_map/include/CGAL/Cell_iterators.h b/Combinatorial_map/include/CGAL/Cell_iterators.h
index 93104ba01e2..9f73b23f232 100644
--- a/Combinatorial_map/include/CGAL/Cell_iterators.h
+++ b/Combinatorial_map/include/CGAL/Cell_iterators.h
@@ -86,7 +86,8 @@ namespace CGAL {
                               Tag_true>::value);
       CGAL_assertion(amap.is_whole_map_unmarked(mcell_mark_number));
 
-      mark_cell<Map,i,dim>(amap, adart, mcell_mark_number);
+      if(this->cont())
+      { mark_cell<Map,i,dim>(amap, adart, mcell_mark_number); }
     }
 
     /// Destructor.
@@ -126,7 +127,8 @@ namespace CGAL {
     {
       unmark_treated_darts();
       Ite::rewind();
-      mark_cell<Map,i,dim>(*this->mmap, (*this), mcell_mark_number);
+      if(this->cont())
+      { mark_cell<Map,i,dim>(*this->mmap, (*this), mcell_mark_number); }
     }
 
     /// Prefix ++ operator.
@@ -199,7 +201,8 @@ namespace CGAL {
       static_assert(std::is_same<typename Ite::Basic_iterator,
                               Tag_true>::value);
       CGAL_assertion(amap.is_whole_map_unmarked(mmark_number));
-      mark_cell<Map,i,dim>(amap, adart, mmark_number);
+      if(this->cont())
+      { mark_cell<Map,i,dim>(amap, adart, mmark_number); }
     }
 
     /// Destructor.
@@ -234,7 +237,8 @@ namespace CGAL {
     {
       unmark_treated_darts();
       Ite::rewind();
-      mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number);
+      if(this->cont())
+      { mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number); }
     }
 
     /// Postfix ++ operator.
@@ -253,7 +257,7 @@ namespace CGAL {
              this->mmap->is_marked((*this), mmark_number));
 
       if (this->cont())
-        mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number);
+      { mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number); }
       return *this;
     }
 
@@ -306,7 +310,8 @@ namespace CGAL {
       static_assert(std::is_same<typename Base::Basic_iterator,
                               Tag_true>::value);
       CGAL_assertion(amap.is_whole_map_unmarked(mmark_number));
-      mark_cell<Map,i,dim>(amap, (*this), mmark_number);
+      if(this->cont())
+      { mark_cell<Map,i,dim>(amap, (*this), mmark_number); }
     }
 
    /// Constructor with a dart in parameter (for end iterator).
@@ -314,8 +319,8 @@ namespace CGAL {
       Base(amap, adart),
       mmark_number(amap.get_new_mark())
     {
-      if (adart!=this->mmap->null_descriptor)
-        mark_cell<Map,i,dim>(amap, (*this), mmark_number);
+      if (this->cont())
+      { mark_cell<Map,i,dim>(amap, (*this), mmark_number); }
     }
 
     /// Destructor.
@@ -350,7 +355,8 @@ namespace CGAL {
     {
       unmark_treated_darts();
       Base::rewind();
-      mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number);
+      if(this->cont())
+      { mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number); }
     }
 
     /// Postfix ++ operator.
@@ -369,7 +375,7 @@ namespace CGAL {
              this->mmap->is_marked((*this), mmark_number));
 
       if (this->cont())
-        mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number);
+      { mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number); }
       return *this;
     }
 
diff --git a/Combinatorial_map/include/CGAL/Compact_container_with_index.h b/Combinatorial_map/include/CGAL/Compact_container_with_index.h
index b14fe21083a..7395743edb0 100644
--- a/Combinatorial_map/include/CGAL/Compact_container_with_index.h
+++ b/Combinatorial_map/include/CGAL/Compact_container_with_index.h
@@ -610,16 +610,16 @@ public:
   }
 
   iterator begin() { return empty()?end():iterator(this, 0, 0); }
-  iterator end()   { return iterator(this, upper_bound()); }
+  iterator end()   { return iterator(this, null_descriptor); }
 
   const_iterator begin() const { return empty()?end():const_iterator(this, 0, 0); }
-  const_iterator end()   const { return const_iterator(this, upper_bound()); }
+  const_iterator end()   const { return const_iterator(this, null_descriptor); }
 
-  reverse_iterator rbegin() { return reverse_iterator(end()); }
+  reverse_iterator rbegin() { return reverse_iterator(iterator(this, upper_bound())); }
   reverse_iterator rend()   { return reverse_iterator(begin()); }
 
   const_reverse_iterator
-  rbegin() const { return const_reverse_iterator(end()); }
+  rbegin() const { return const_reverse_iterator(iterator(this, upper_bound())); }
   const_reverse_iterator
   rend()   const { return const_reverse_iterator(begin()); }
 
@@ -878,7 +878,8 @@ namespace internal {
     CC_iterator_with_index(const iterator &it): m_ptr_to_cc(it.m_ptr_to_cc),
       m_index(it.m_index)
     {
-      CGAL_assertion(m_index<=m_ptr_to_cc->upper_bound());
+       CGAL_assertion(m_index<=m_ptr_to_cc->upper_bound() ||
+                     m_index==DSC::null_descriptor);
     }
 
     // Same for assignment operator
@@ -928,14 +929,16 @@ namespace internal {
       // It's either pointing to end(), or valid.
       CGAL_assertion_msg(m_ptr_to_cc!=nullptr,
          "Incrementing a singular iterator or an empty container iterator ?");
-      CGAL_assertion_msg(m_index<m_ptr_to_cc->upper_bound(),
-         "Incrementing end() ?");
+      CGAL_assertion_msg(m_index<m_ptr_to_cc->upper_bound() &&
+                             m_index!=DSC::null_descriptor,
+                         "Incrementing end() ?");
 
       // If it's not end(), then it's valid, we can do ++.
       do
       { ++m_index; }
       while(m_index<m_ptr_to_cc->upper_bound() &&
             (!m_ptr_to_cc->is_used(m_index)));
+      if(m_index==m_ptr_to_cc->upper_bound()) { m_index=DSC::null_descriptor; }
     }
 
     void decrement()
@@ -969,7 +972,7 @@ namespace internal {
 
     pointer   operator->() const { return &((*m_ptr_to_cc)[m_index]); }
 
-    bool is_end() const { return m_index>=m_ptr_to_cc->upper_bound(); }
+    bool is_end() const { return m_index==DSC::null_descriptor; }
 
     // Can itself be used for bit-squatting.
     size_type for_compact_container() const
diff --git a/Generalized_map/include/CGAL/GMap_cell_iterators.h b/Generalized_map/include/CGAL/GMap_cell_iterators.h
index 1dbef9dcd6e..d739d64d3c2 100644
--- a/Generalized_map/include/CGAL/GMap_cell_iterators.h
+++ b/Generalized_map/include/CGAL/GMap_cell_iterators.h
@@ -73,7 +73,8 @@ namespace CGAL {
       static_assert(std::is_same<typename Base::Basic_iterator,
                               Tag_true>::value);
       CGAL_assertion(amap.is_whole_map_unmarked(mmark_number));
-      mark_cell<Map,i,dim>(amap, (*this), mmark_number);
+      if(this->cont())
+      { mark_cell<Map,i,dim>(amap, (*this), mmark_number); }
     }
 
    /// Constructor with a dart in parameter (for end iterator).
@@ -81,8 +82,8 @@ namespace CGAL {
       Base(amap, adart),
       mmark_number(amap.get_new_mark())
     {
-      if (adart!=this->mmap->null_descriptor)
-        mark_cell<Map,i,dim>(amap, (*this), mmark_number);
+      if (this->cont())
+      { mark_cell<Map,i,dim>(amap, (*this), mmark_number); }
     }
 
     /// Destructor.
@@ -136,7 +137,7 @@ namespace CGAL {
              this->mmap->is_marked((*this), mmark_number));
 
       if (this->cont())
-        mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number);
+      { mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number); }
       return *this;
     }
 
@@ -249,7 +250,7 @@ namespace CGAL {
              this->mmap->is_marked((*this), mmark_number));
 
       if (this->cont())
-        mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number);
+      { mark_cell<Map,i,dim>(*this->mmap, (*this), mmark_number); }
       return *this;
     }