diff --git a/Constrained_triangulation_3/examples/Constrained_triangulation_3/remesh_constrained_Delaunay_triangulation_3.cpp b/Constrained_triangulation_3/examples/Constrained_triangulation_3/remesh_constrained_Delaunay_triangulation_3.cpp
index 490a284fb48..956ed10fdf6 100644
--- a/Constrained_triangulation_3/examples/Constrained_triangulation_3/remesh_constrained_Delaunay_triangulation_3.cpp
+++ b/Constrained_triangulation_3/examples/Constrained_triangulation_3/remesh_constrained_Delaunay_triangulation_3.cpp
@@ -11,11 +11,10 @@
 #include <CGAL/draw_triangulation_3.h>
 
 using K = CGAL::Exact_predicates_inexact_constructions_kernel;
-using Vb = CGAL::Constrained_Delaunay_triangulation_vertex_base_3<K, CGAL::Tetrahedral_remeshing::Remeshing_vertex_base_3<K>>;
-using Cb = CGAL::Constrained_Delaunay_triangulation_cell_base_3<K, CGAL::Tetrahedral_remeshing::Remeshing_cell_base_3<K>>;
+using Vb = CGAL::Constrained_Delaunay_triangulation_vertex_base_3<K>;
+using Cb = CGAL::Constrained_Delaunay_triangulation_cell_base_3<K>;
 using Tds = CGAL::Triangulation_data_structure_3<Vb, Cb>;
 using Tr = CGAL::Triangulation_3<K, Tds>;
-using CDt = CGAL::Constrained_Delaunay_triangulation_3<K, Tr>;
 
 int main(int argc, char* argv[])
 {
diff --git a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h
index a40c78469f1..1bb3988f2a1 100644
--- a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h
+++ b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h
@@ -26,8 +26,8 @@
 #include <CGAL/Constrained_triangulation_3/internal/config.h>
 
 #include <CGAL/Base_with_time_stamp.h>
-#include <CGAL/Constrained_Delaunay_triangulation_cell_data_3.h>
 #include <CGAL/Constrained_Delaunay_triangulation_vertex_base_3.h>
+#include <CGAL/Constrained_Delaunay_triangulation_cell_base_3.h>
 #include <CGAL/Triangulation_vertex_base_3.h>
 #include <CGAL/Triangulation_cell_base_3.h>
 #include <CGAL/Triangulation_vertex_base_with_info_2.h>
@@ -74,6 +74,7 @@
 
 #ifndef DOXYGEN_RUNNING
 
+#if 0
 namespace CGAL {
 
   /*!
@@ -170,6 +171,7 @@ namespace CGAL {
   };
 
 } // end namespace CGAL
+#endif // 0
 
 namespace CGAL {
 
@@ -524,70 +526,6 @@ concept Range_of_polygon_3 = std::ranges::common_range<Polygons>
       && Polygon_3<std::ranges::range_value_t<Polygons>, Kernel>;
 #endif // concepts
 
-template <typename Gt, typename Cb = Triangulation_cell_base_3<Gt> >
-class Constrained_Delaunay_triangulation_cell_base_3
-  : public Base_with_time_stamp<Cb>
-{
-  using Base = Base_with_time_stamp<Cb>;
-  Constrained_Delaunay_triangulation_cell_data_3 cdt_3_data_;
-public:
-  // To get correct cell type in TDS
-  template < class TDS3 >
-  struct Rebind_TDS {
-    typedef typename Cb::template Rebind_TDS<TDS3>::Other Cb3;
-    typedef Constrained_Delaunay_triangulation_cell_base_3 <Gt, Cb3> Other;
-  };
-
-  // Constructor
-  using Base::Base;
-
-  Constrained_Delaunay_triangulation_cell_data_3& cdt_3_data() {
-    return cdt_3_data_;
-  }
-
-  const Constrained_Delaunay_triangulation_cell_data_3& cdt_3_data() const {
-    return cdt_3_data_;
-  }
-
-  static std::string io_signature() {
-    return Get_io_signature<Base>()() + "+(" + Get_io_signature<int>()()
-      + ")[4]";
-  }
-
-  friend std::ostream&
-  operator<<(std::ostream& os,
-             const Constrained_Delaunay_triangulation_cell_base_3& c)
-  {
-    os << static_cast<const Base&>(c);
-    for( unsigned li = 0; li < 4; ++li ) {
-      if(IO::is_ascii(os)) {
-        os << " " << c.cdt_3_data().face_id[li];
-      } else {
-        CGAL::write(os, c.cdt_3_data().face_id[li]);
-      }
-    }
-    return os;
-  }
-  friend std::istream&
-  operator>>(std::istream& is,
-             Constrained_Delaunay_triangulation_cell_base_3& c)
-  {
-    is >> static_cast<Base&>(c);
-    if(!is) return is;
-    for( int li = 0; li < 4; ++li ) {
-      int i;
-      if(IO::is_ascii(is)) {
-        is >> i;
-      } else {
-        CGAL::read(is, i);
-      }
-      if(!is) return is;
-      c.face_id[li] = i;
-    }
-    return is;
-  }
-};
-
 template <class DSC, bool Const>
 struct Output_rep<CGAL::internal::CC_iterator<DSC, Const>, With_point_and_info_tag>
   : public Output_rep<CGAL::internal::CC_iterator<DSC, Const>>
diff --git a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_base_3.h b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_base_3.h
index 2ea46918955..78a8a967c0d 100644
--- a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_base_3.h
+++ b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_base_3.h
@@ -21,6 +21,7 @@
 #ifndef CGAL_CONSTRAINED_DELAUNAY_TRIANGULATION_CELL_BASE_3_H
 #define CGAL_CONSTRAINED_DELAUNAY_TRIANGULATION_CELL_BASE_3_H
 
+#include <CGAL/Constrained_Delaunay_triangulation_cell_data_3.h>
 #include <CGAL/Triangulation_cell_base_3.h>
 
 namespace CGAL {
@@ -43,9 +44,93 @@ namespace CGAL {
  *
  * \sa `CGAL::Constrained_Delaunay_triangulation_vertex_base_3`
  */
-template < typename Gt, typename Cb = Triangulation_cell_base_3<Gt> >
-class Constrained_Delaunay_triangulation_cell_base_3 : public Cb {
+template <typename Gt, typename Cb = Triangulation_cell_base_3<Gt> >
+class Constrained_Delaunay_triangulation_cell_base_3
+  : public Base_with_time_stamp<Cb>
+{
+  using Base = Base_with_time_stamp<Cb>;
+  Constrained_Delaunay_triangulation_cell_data_3 cdt_3_data_;
+
+  mutable bool sliver_cache_validity_ = false;
+  CDT_3_face_index subdomain_index_ = -1;
+  double sliver_value_ = 0.;
 public:
+  // To get correct cell type in TDS
+  template < class TDS3 >
+  struct Rebind_TDS {
+    typedef typename Cb::template Rebind_TDS<TDS3>::Other Cb3;
+    typedef Constrained_Delaunay_triangulation_cell_base_3 <Gt, Cb3> Other;
+  };
+
+  // Constructors inherited from the base class
+  using Base::Base;
+
+  Constrained_Delaunay_triangulation_cell_data_3& cdt_3_data() {
+    return cdt_3_data_;
+  }
+
+  const Constrained_Delaunay_triangulation_cell_data_3& cdt_3_data() const {
+    return cdt_3_data_;
+  }
+
+  // model of SimplicialMeshCellBase_3
+  using Surface_patch_index = CDT_3_face_index;
+  using Subdomain_index = CDT_3_face_index;
+  bool is_facet_on_surface(int i) const { return cdt_3_data().is_facet_constrained(i); }
+  Surface_patch_index surface_patch_index(int i) const { return cdt_3_data().face_constraint_index(i) + 1; }
+  void set_surface_patch_index(int i, Surface_patch_index index) { cdt_3_data().face_id[unsigned(i)] = index - 1; }
+  Subdomain_index subdomain_index() const { return subdomain_index_; }
+  void set_subdomain_index(Subdomain_index i) { subdomain_index_ = i; }
+
+  // model of RemeshingCellBase_3
+  void set_sliver_value(double value) {
+    sliver_cache_validity_ = true;
+    sliver_value_ = value;
+  }
+  double sliver_value() const {
+    CGAL_assertion(is_cache_valid());
+    return sliver_value_;
+  }
+  bool is_cache_valid() const { return sliver_cache_validity_; }
+  void reset_cache_validity() const { sliver_cache_validity_ = false; }
+
+  static std::string io_signature() {
+    return Get_io_signature<Base>()() + "+(" + Get_io_signature<int>()()
+      + ")[4]";
+  }
+
+  friend std::ostream&
+  operator<<(std::ostream& os,
+             const Constrained_Delaunay_triangulation_cell_base_3& c)
+  {
+    os << static_cast<const Base&>(c);
+    for( unsigned li = 0; li < 4; ++li ) {
+      if(IO::is_ascii(os)) {
+        os << " " << c.cdt_3_data().face_id[li];
+      } else {
+        CGAL::write(os, c.cdt_3_data().face_id[li]);
+      }
+    }
+    return os;
+  }
+  friend std::istream&
+  operator>>(std::istream& is,
+             Constrained_Delaunay_triangulation_cell_base_3& c)
+  {
+    is >> static_cast<Base&>(c);
+    if(!is) return is;
+    for( int li = 0; li < 4; ++li ) {
+      int i;
+      if(IO::is_ascii(is)) {
+        is >> i;
+      } else {
+        CGAL::read(is, i);
+      }
+      if(!is) return is;
+      c.face_id[li] = i;
+    }
+    return is;
+  }
 };
 
 } // namespace CGAL
diff --git a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_vertex_base_3.h b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_vertex_base_3.h
index c700e5e9408..70a28e29683 100644
--- a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_vertex_base_3.h
+++ b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_vertex_base_3.h
@@ -48,6 +48,12 @@ template < typename Gt, typename Vb = Triangulation_vertex_base_3<Gt> >
 class Constrained_Delaunay_triangulation_vertex_base_3 : public Base_with_time_stamp<Vb>
 {
   Constrained_Delaunay_triangulation_vertex_data_3 cdt_3_data_;
+  bool cache_validity_ = false;
+  CDT_3_face_index index_ = 0;
+  int dim_ = -1;
+  std::size_t number_of_incident_facets_ = 0;
+  std::size_t number_of_components_ = 0;
+
 public:
   // To get correct vertex type in TDS
   template <class TDS3> struct Rebind_TDS
@@ -56,16 +62,54 @@ public:
     using Other = Constrained_Delaunay_triangulation_vertex_base_3<Gt, Vb3>;
   };
 
+  // constructors, inherited from the base class
   using Base = Base_with_time_stamp<Vb>;
   using Base::Base;
 
-  Constrained_Delaunay_triangulation_vertex_data_3& cdt_3_data() {
-    return cdt_3_data_;
+  // model of SimplicialMeshVertexBase_3
+  using Index = CDT_3_face_index;
+  int in_dimension() const { return dim_; }
+  void set_dimension(int d) { dim_ = d; }
+  Index index() const { return index_; }
+  void set_index(Index i) { index_ = i; }
+  bool is_c2t3_cache_valid() const { return cache_validity_; }
+  void invalidate_c2t3_cache() { cache_validity_ = false; }
+  void set_c2t3_cache(std::size_t i, std::size_t j)
+  {
+    number_of_incident_facets_ = i;
+    number_of_components_ = j;
+    cache_validity_ = true;
+  }
+  std::size_t cached_number_of_incident_facets() const { return number_of_incident_facets_; }
+  std::size_t cached_number_of_components() const { return number_of_components_; }
+
+  void sync() {
+    switch(cdt_3_data().vertex_type()) {
+      case CDT_3_vertex_type::FREE:
+        set_dimension(3);
+        set_index(0);
+        break;
+      case CDT_3_vertex_type::CORNER:
+        set_dimension(0);
+        set_index(0);
+        break;
+      case CDT_3_vertex_type::STEINER_ON_EDGE:
+        set_dimension(1);
+        set_index(0);
+        break;
+      case CDT_3_vertex_type::STEINER_IN_FACE:
+        set_dimension(2);
+        set_index(cdt_3_data().face_index());
+        break;
+      default:
+        CGAL_error();
+        break;
+    }
   }
 
-  const Constrained_Delaunay_triangulation_vertex_data_3& cdt_3_data() const {
-    return cdt_3_data_;
-  }
+  // model of ConstrainedDelaunayTriangulationVertexBase_3
+  Constrained_Delaunay_triangulation_vertex_data_3& cdt_3_data() { return cdt_3_data_; }
+  const Constrained_Delaunay_triangulation_vertex_data_3& cdt_3_data() const { return cdt_3_data_; }
 
   static std::string io_signature() {
     return Get_io_signature<Vb>()();
diff --git a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_vertex_data_3.h b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_vertex_data_3.h
index 30bdde4b4f1..ac37e0738a8 100644
--- a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_vertex_data_3.h
+++ b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_vertex_data_3.h
@@ -25,6 +25,8 @@
 
 #include <CGAL/Constrained_triangulation_3/internal/config.h>
 
+#include <bitset>
+
 namespace CGAL {
 
 #ifdef DOXYGEN_RUNNING
@@ -53,6 +55,7 @@ enum class CDT_3_vertex_marker {
 
 struct Constrained_Delaunay_triangulation_vertex_data_3 {
 protected:
+  // TODO: check and improve the compactness of this class
   CDT_3_vertex_type m_vertex_type = CDT_3_vertex_type::FREE;
   std::bitset<static_cast<int>(CDT_3_vertex_marker::nb_of_markers)> mark{};
   union U {
diff --git a/Constrained_triangulation_3/include/CGAL/make_constrained_Delaunay_triangulation_3.h b/Constrained_triangulation_3/include/CGAL/make_constrained_Delaunay_triangulation_3.h
index a0ab8fdce5d..6044e06f910 100644
--- a/Constrained_triangulation_3/include/CGAL/make_constrained_Delaunay_triangulation_3.h
+++ b/Constrained_triangulation_3/include/CGAL/make_constrained_Delaunay_triangulation_3.h
@@ -86,7 +86,32 @@ Triangulation_3
 make_constrained_Delaunay_triangulation_3(const PolygonMesh& mesh, const NamedParams& np = parameters::default_values())
 {
   Constrained_Delaunay_triangulation_3<typename Triangulation_3::Geom_traits, Triangulation_3> cdt(mesh, np);
-  return std::move(cdt).triangulation();
+  Triangulation_3 tr = std::move(cdt).triangulation();
+
+  for(auto vh : tr.all_vertex_handles()) {
+    vh->sync();
+  }
+
+  for(auto ch : tr.all_cell_handles()) {
+    ch->set_subdomain_index(1);
+  }
+
+  std::stack<typename Triangulation_3::Cell_handle> stack;
+  stack.push(tr.infinite_cell());
+  while(!stack.empty()) {
+    auto ch = stack.top();
+    stack.pop();
+    ch->set_subdomain_index(0);
+    for(int i = 0; i < 4; ++i) {
+      if(ch->is_facet_on_surface(i)) continue;
+      auto n = ch->neighbor(i);
+      if(n->subdomain_index() == 1) {
+        stack.push(n);
+      }
+    }
+  }
+
+  return tr;
 }
 
 } // end namespace CGAL