diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_region.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_region.h
index 9a7f8111fa7..5d6ae21f8c7 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_region.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_region.h
@@ -66,7 +66,16 @@ namespace Polygon_mesh {
     using Face_graph = PolygonMesh;
     using Face_range = FaceRange;
     using Vertex_to_point_map = VertexToPointMap;
-    using Face_to_region_map = internal::Item_to_region_index_map;
+
+    using Face_to_index_map = internal::Item_to_index_property_map<Face_range>;
+    using Face_to_region_index_map = internal::Item_to_region_index_map<Face_to_index_map>;
+
+    struct Face_to_region_map : public Face_to_region_index_map {
+      template<typename Regions>
+      Face_to_region_map(const Face_range& face_range, const Regions& regions) :
+      Face_to_region_index_map(face_range, Face_to_index_map(face_range), regions)
+      { }
+    };
     /// \endcond
 
     /// Number type.
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Polyline_graph.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Polyline_graph.h
index 9acf46d960b..72963983a79 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Polyline_graph.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Polyline_graph.h
@@ -69,7 +69,7 @@ namespace Polygon_mesh {
       std::size_t index = std::size_t(-1);
       std::set<std::size_t> sneighbors;
       std::set<std::size_t> tneighbors;
-      std::pair<int, int> regions;
+      std::pair<long, long> regions;
     };
 
   public:
@@ -179,7 +179,7 @@ namespace Polygon_mesh {
     void build_graph() {
 
       clear();
-      int r1 = -1, r2 = -1;
+      long r1 = -1, r2 = -1;
       std::vector<std::size_t> pedge_map(
         m_edge_range.size(), std::size_t(-1));
       for (const auto& edge : m_edge_range) {
@@ -301,7 +301,7 @@ namespace Polygon_mesh {
 
       \pre `query_index < segment_range().size()`
     */
-    const std::pair<int, int>& edge_regions(
+    const std::pair<long, long>& edge_regions(
       const std::size_t query_index) const {
 
       CGAL_precondition(query_index < segment_range().size());
@@ -334,7 +334,7 @@ namespace Polygon_mesh {
     std::vector<PEdge> m_pedges;
 
     template<typename EdgeType>
-    std::pair<int, int> get_regions(const EdgeType& edge) const {
+    std::pair<long, long> get_regions(const EdgeType& edge) const {
 
       const auto hedge1 = halfedge(edge, m_face_graph);
       const auto hedge2 = opposite(hedge1, m_face_graph);
@@ -346,17 +346,17 @@ namespace Polygon_mesh {
       const std::size_t fi2 = get(m_face_to_index_map, face2);
       CGAL_precondition(fi1 != fi2);
 
-      int r1 = -1, r2 = -1;
+      long r1 = -1, r2 = -1;
       if (fi1 != std::size_t(-1))
-        r1 = get(m_face_to_region_map, fi1);
+        r1 = get(m_face_to_region_map, face1);
       if (fi2 != std::size_t(-1))
-        r2 = get(m_face_to_region_map, fi2);
+        r2 = get(m_face_to_region_map, face2);
       return std::make_pair(r1, r2);
     }
 
     template<typename EdgeType>
     void add_graph_edge(
-      const EdgeType& edge, const int region1, const int region2,
+      const EdgeType& edge, const long region1, const long region2,
       std::vector<std::size_t>& pedge_map) {
 
       PEdge pedge;
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/property_map.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/property_map.h
index cb9f8f572ce..1d06f8b6c73 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/property_map.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/property_map.h
@@ -42,17 +42,14 @@ namespace internal {
   class Item_property_map {
 
   public:
-    using Item_range = ItemRange;
-    using Property_map = PropertyMap;
-
-    using value_type = typename Property_map::value_type;
-    using reference = const value_type&;
     using key_type = std::size_t;
+    using value_type = typename PropertyMap::value_type;
+    using reference = const value_type&;
     using category = boost::lvalue_property_map_tag;
 
     Item_property_map(
-      const Item_range& item_range,
-      const Property_map& property_map) :
+      const ItemRange& item_range,
+      const PropertyMap& property_map) :
     m_item_range(item_range),
     m_property_map(property_map)
     { }
@@ -69,26 +66,20 @@ namespace internal {
     }
 
   private:
-    const Item_range& m_item_range;
-    const Property_map& m_property_map;
+    const ItemRange& m_item_range;
+    const PropertyMap& m_property_map;
   };
 
   template<typename ItemRange>
   class Item_to_index_property_map {
 
   public:
-    using Item_range = ItemRange;
-
-    using Iterator = typename Item_range::const_iterator;
-    using Item = typename Iterator::value_type;
-
+    using key_type = typename ItemRange::const_iterator::value_type;
     using value_type = std::size_t;
-    using key_type = Item;
+    using reference = value_type;
     using category = boost::readable_property_map_tag;
 
-    using Item_map = std::map<key_type, value_type>;
-
-    Item_to_index_property_map(const Item_range& item_range) :
+    Item_to_index_property_map(const ItemRange& item_range) :
     m_item_range(item_range) {
 
       value_type i = 0;
@@ -112,8 +103,8 @@ namespace internal {
     }
 
   private:
-    const Item_range& m_item_range;
-    Item_map m_item_map;
+    const ItemRange& m_item_range;
+    std::map<key_type, value_type> m_item_map;
   };
 
   class Seed_property_map {
@@ -121,6 +112,7 @@ namespace internal {
   public:
     using key_type = std::size_t;
     using value_type = std::size_t;
+    using reference = value_type;
     using category = boost::readable_property_map_tag;
 
     Seed_property_map(
@@ -141,41 +133,52 @@ namespace internal {
     const std::vector<std::size_t>& m_seeds;
   };
 
+  template<typename ItemToIndexMap>
   class Item_to_region_index_map {
 
   public:
-    using key_type = std::size_t;
-    using value_type = int;
+    using key_type = typename ItemToIndexMap::key_type;
+    using value_type = long;
     using reference = value_type;
     using category = boost::readable_property_map_tag;
 
-    Item_to_region_index_map() { }
+    Item_to_region_index_map() :
+    m_item_to_index_map(nullptr)
+    { }
 
-    template<typename ItemRange>
+    template<typename ItemRange, typename RegionRange>
     Item_to_region_index_map(
       const ItemRange& items,
-      const std::vector< std::vector<std::size_t> >& regions) :
+      const ItemToIndexMap& item_to_index_map,
+      const RegionRange& regions) :
+    m_item_to_index_map(std::make_shared<ItemToIndexMap>(item_to_index_map)),
     m_indices(items.size(), -1) {
 
-      for (std::size_t i = 0; i < regions.size(); ++i) {
-        for (const std::size_t index : regions[i]) {
+      long region_index = 0;
+      for (const auto& region : regions) {
+        for (const auto index : region) {
           CGAL_precondition(index < m_indices.size());
-          m_indices[index] = static_cast<int>(i);
+          m_indices[index] = region_index;
         }
+        ++region_index;
       }
+      CGAL_precondition(region_index == static_cast<long>(regions.size()));
     }
 
     inline friend value_type get(
       const Item_to_region_index_map& item_to_region_index_map,
-      const key_type key) {
+      const key_type& key) {
 
       const auto& indices = item_to_region_index_map.m_indices;
-      CGAL_precondition(key < indices.size());
-      return indices[key];
+      const auto& item_to_index_map = item_to_region_index_map.m_item_to_index_map;
+      const std::size_t item_index = get(*item_to_index_map, key);
+      CGAL_precondition(item_index < indices.size());
+      return indices[item_index];
     }
 
   private:
-    std::vector<int> m_indices;
+    const std::shared_ptr<ItemToIndexMap> m_item_to_index_map;
+    std::vector<long> m_indices;
   };
 
 } // namespace internal
diff --git a/Shape_detection/todo.md b/Shape_detection/todo.md
index 4a28799d53b..baea6ec34ed 100644
--- a/Shape_detection/todo.md
+++ b/Shape_detection/todo.md
@@ -1,4 +1,3 @@
-- make a better version of the Polyline_graph where I return a filtered set of original edges instead of PEdge
 - update the polyhedron demo
 - ---- submission ----
 - add new tests: polyline (2D, 3D), polyline with sorting (2D, 3D), free functions, randomness, strict tests on the data from PMP, tests similar to the basic_example, check polylines with equal points