diff --git a/Polygon_mesh_processing/doc/Polygon_mesh_processing/Concepts/PMPOrientationVisitor.h b/Polygon_mesh_processing/doc/Polygon_mesh_processing/Concepts/PMPOrientationVisitor.h
index e1a9797f9c6..c0b20d83c34 100644
--- a/Polygon_mesh_processing/doc/Polygon_mesh_processing/Concepts/PMPOrientationVisitor.h
+++ b/Polygon_mesh_processing/doc/Polygon_mesh_processing/Concepts/PMPOrientationVisitor.h
@@ -17,11 +17,23 @@ public:
 
 /// called each time an edge appears in more than two polygons.
 /// `id1` and `id2` are the vertex ids of the endpoints of the edge.
-void non_manifold_edge(std::size_t id1, std::size_t id2);
+/// `nb_polygons` indicates the number of polygons containing that edge.
+void non_manifold_edge(std::size_t id1, std::size_t id2, std::size_t nb_polygons);
 
 /// called each time a non-manifold vertex is detected.
-/// `id` is the id of the vertex has is non-manifold.
-void non_manifold_vertex(std::size_t id);
+/// `vid` is the id of the vertex that is non-manifold.
+/// `nb_link_ccs` is the number of edge connected components of vertices
+/// in the link of the corresponding vertex.
+void non_manifold_vertex(std::size_t vid, std::size_t nb_link_ccs);
+
+/// called during the detection of a non-manifold vertex, one time
+/// per incident edge connected component of vertices in the link of the vertex with id `vid`.
+/// `id` is the id of the vertex that is non-manifold.
+/// `polygon_ids` contains the ids of the polygon in such a connected component.
+/// This function is called a number of times exactly equal to the parameter `nb_link_ccs`
+/// for the same vertex in `non_manifold_vertex()`. Note that the aforementioned function is
+/// called after all the calls to this function are done.
+void link_connected_polygons(std::size_t vid, const std::vector<std::size_t>& polygon_ids){}
 
 /// @}
 
diff --git a/Polygon_mesh_processing/examples/Polygon_mesh_processing/orient_polygon_soup_example.cpp b/Polygon_mesh_processing/examples/Polygon_mesh_processing/orient_polygon_soup_example.cpp
index 7416b4aabbd..67b13a79a8b 100644
--- a/Polygon_mesh_processing/examples/Polygon_mesh_processing/orient_polygon_soup_example.cpp
+++ b/Polygon_mesh_processing/examples/Polygon_mesh_processing/orient_polygon_soup_example.cpp
@@ -16,26 +16,27 @@
 typedef CGAL::Exact_predicates_inexact_constructions_kernel          K;
 typedef CGAL::Polyhedron_3<K, CGAL::Polyhedron_items_with_id_3>      Polyhedron;
 
-//Optional visitor for orientation to demonstrate usage
+// Optional visitor for orientating a polygon soup to demonstrate usage for some functions.
+// inherits from the default class as some functions are not overloaded
 struct Visitor : public CGAL::Polygon_mesh_processing::Default_orientation_visitor
 {
-  void non_manifold_edge(std::size_t id1, std::size_t id2) final
+  void non_manifold_edge(std::size_t id1, std::size_t id2, std::size_t nb_poly)
   {
-    std::cout << "The edge " << id1 << ", " << id2 << " is not manifold." << std::endl;
+    std::cout << "The edge " << id1 << ", " << id2 << " is not manifold: " << nb_poly << " incident polygons." << std::endl;
   }
-  void non_manifold_vertex(const std::size_t & id) final
+  void non_manifold_vertex(std::size_t id, std::size_t nb_cycles)
   {
-     std::cout << "The vertex " << id << " is not manifold." << std::endl;
+     std::cout << "The vertex " << id << " is not manifold: "  << nb_lcc<< " connected components of vertices in the link." << std::endl;
   }
-  void duplicated_vertex(std::size_t v1, std::size_t v2) final
+  void duplicated_vertex(std::size_t v1, std::size_t v2)
   {
     std::cout << "The vertex " << v1 << " has been duplicated, its new id is " << v2 << "." << std::endl;
   }
-  void vertex_id_in_polygon_replaced(std::size_t p_id, std::size_t i1, std::size_t i2) final
+  void vertex_id_in_polygon_replaced(std::size_t p_id, std::size_t i1, std::size_t i2)
   {
     std::cout << "In the polygon " << p_id << ", the index " << i1 << " has been replaced by " << i2 << "." << std::endl;
   }
-  void polygon_orientation_reversed(std::size_t p_id) final
+  void polygon_orientation_reversed(std::size_t p_id)
   {
     std::cout << "The polygon " << p_id << " has been reversed." << std::endl;
   }
diff --git a/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/orient_polygon_soup.h b/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/orient_polygon_soup.h
index 39ad97d39ea..bcc364ea1a3 100644
--- a/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/orient_polygon_soup.h
+++ b/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/orient_polygon_soup.h
@@ -45,11 +45,12 @@ namespace Polygon_mesh_processing {
  *  overridden.
  */
 struct Default_orientation_visitor{
-  inline virtual void non_manifold_edge(std::size_t, std::size_t){}
-  inline virtual void non_manifold_vertex(std::size_t){}
-  inline virtual void duplicated_vertex(std::size_t, std::size_t){}
-  inline virtual void vertex_id_in_polygon_replaced(std::size_t, std::size_t, std::size_t){}
-  inline virtual void polygon_orientation_reversed(std::size_t) {}
+  void non_manifold_edge(std::size_t, std::size_t, std::size_t){}
+  void non_manifold_vertex(std::size_t,std::size_t){}
+  void duplicated_vertex(std::size_t, std::size_t){}
+  void vertex_id_in_polygon_replaced(std::size_t, std::size_t, std::size_t){}
+  void polygon_orientation_reversed(std::size_t) {}
+  void link_connected_polygons(std::size_t, const std::vector<std::size_t>&){}
 };
 
 namespace internal {
@@ -80,7 +81,7 @@ struct Polygon_soup_orienter
   Edge_map edges;             //< the set of edges of the input polygons
   Marked_edges marked_edges;  //< the set of singular edges or edges incident
                               //<   to non-compatible orientation polygons
-  Visitor visitor;
+  Visitor& visitor;
 
   /// for each polygon referenced by its position in `polygons`, indicates
   /// the connected component it belongs too after orientation.
@@ -168,18 +169,12 @@ struct Polygon_soup_orienter
     }
   }
 
-  Polygon_soup_orienter(Points& points, Polygons& polygons, Visitor visitor)
+  Polygon_soup_orienter(Points& points, Polygons& polygons, Visitor& visitor)
     : points(points), polygons(polygons), edges(points.size()), visitor(visitor)
   {}
 
-  Polygon_soup_orienter(Points& points, Polygons& polygons)
-    : points(points), polygons(polygons), edges(points.size())
-  {
-  }
-
-
 //filling containers
-  static void fill_edge_map(Edge_map& edges, Marked_edges& marked_edges, const Polygons& polygons, Visitor visitor) {
+  static void fill_edge_map(Edge_map& edges, Marked_edges& marked_edges, const Polygons& polygons, Visitor& visitor) {
     // Fill edges
     for (P_ID i = 0; i < polygons.size(); ++i)
     {
@@ -208,7 +203,7 @@ struct Polygon_soup_orienter
 
         if (nb_edges > 2)
         {
-          visitor.non_manifold_edge(i0, i1);
+          visitor.non_manifold_edge(i0, i1, nb_edges);
           set_edge_marked(i0, i1, marked_edges);
         }
       }
@@ -357,24 +352,31 @@ struct Polygon_soup_orienter
       if ( incident_polygons.empty() ) continue; //isolated vertex
       std::set<P_ID> visited_polygons;
 
-      bool first_pass = true;
-      bool is_nm = false;
+      std::size_t nb_link_ccs=0;
       for(P_ID p_id : incident_polygons)
       {
         if ( !visited_polygons.insert(p_id).second ) continue; // already visited
 
-        if (!first_pass)
+        if (++nb_link_ccs != 1)
         {
+          // the vertex is non-manifold
           vertices_to_duplicate.push_back(std::pair<V_ID, std::vector<P_ID> >());
           vertices_to_duplicate.back().first=v_id;
-          is_nm = true;
+
+          // call the visitor only if the function has been overridden
+          if (nb_link_ccs==2 && &Visitor::link_connected_polygons != &Default_orientation_visitor::link_connected_polygons)
+          {
+            std::vector<std::size_t> tmp(visited_polygons.begin(), visited_polygons.end());
+            tmp.erase(std::remove(tmp.begin(), tmp.end(), p_id), tmp.end());
+            visitor.link_connected_polygons(v_id, tmp);
+          }
         }
 
         const std::array<V_ID,3>& neighbors = get_neighbor_vertices(v_id,p_id,polygons);
 
         V_ID next = neighbors[2];
 
-        if( !first_pass)
+        if(nb_link_ccs != 1)
           vertices_to_duplicate.back().second.push_back(p_id);
 
         do{
@@ -382,7 +384,7 @@ struct Polygon_soup_orienter
           std::tie(next, other_p_id) = next_cw_vertex_around_source(v_id, next, polygons, edges, marked_edges);
           if (next==v_id) break;
           visited_polygons.insert(other_p_id);
-          if( !first_pass)
+          if(nb_link_ccs != 1)
             vertices_to_duplicate.back().second.push_back(other_p_id);
         }
         while(next!=neighbors[0]);
@@ -395,17 +397,15 @@ struct Polygon_soup_orienter
             std::tie(next, other_p_id) = next_ccw_vertex_around_target(next, v_id, polygons, edges, marked_edges);
             if (next==v_id) break;
             visited_polygons.insert(other_p_id);
-            if( !first_pass)
+            if(nb_link_ccs != 1)
               vertices_to_duplicate.back().second.push_back(other_p_id);
           }
           while(true);
         }
-        first_pass=false;
-      }
-      if (is_nm)
-      {
-        visitor.non_manifold_vertex(v_id);
+        if (nb_link_ccs != 1)
+          visitor.link_connected_polygons(v_id, vertices_to_duplicate.back().second);
       }
+      if (nb_link_ccs != 1) visitor.non_manifold_vertex(v_id, nb_link_ccs);
     }
 
     /// now duplicate the vertices
diff --git a/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/orient_polygon_soup_extension.h b/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/orient_polygon_soup_extension.h
index 91712ccfda8..2a9b8e05099 100644
--- a/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/orient_polygon_soup_extension.h
+++ b/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/orient_polygon_soup_extension.h
@@ -64,7 +64,8 @@ duplicate_non_manifold_edges_in_polygon_soup(PointRange& points,
   typedef CGAL::Polygon_mesh_processing::internal::
     Polygon_soup_orienter<PointRange, PolygonRange> Orienter;
 
-  Orienter orienter(points, polygons);
+  Default_orientation_visitor visitor;
+  Orienter orienter(points, polygons, visitor);
   orienter.fill_edge_map();
   // make edges to duplicate
   for(std::size_t i1=0;i1<points.size();++i1)
diff --git a/Polyhedron/demo/Polyhedron/Plugins/PMP/Orient_soup_plugin.cpp b/Polyhedron/demo/Polyhedron/Plugins/PMP/Orient_soup_plugin.cpp
index afa8f4b695b..7ef5a3f2e58 100644
--- a/Polyhedron/demo/Polyhedron/Plugins/PMP/Orient_soup_plugin.cpp
+++ b/Polyhedron/demo/Polyhedron/Plugins/PMP/Orient_soup_plugin.cpp
@@ -254,25 +254,8 @@ void Polyhedron_demo_orient_soup_plugin::createPointsAndPolyline(std::vector<std
     QApplication::setOverrideCursor(Qt::WaitCursor);
     Scene_points_with_normal_item* points = nullptr;
 
-    //remove vertices already in NM edges
-
-    std::unordered_set<std::size_t> check_set;
-    for(Scene_polygon_soup_item::Edge edge : item->non_manifold_edges())
-    {
-      check_set.insert(edge[0]);
-      check_set.insert(edge[1]);
-    }
-
-    std::set<std::size_t> nm_vertices;
-    for(const auto& v : nm_points)
-    {
-      if(check_set.insert(v).second)
-      {
-        nm_vertices.insert(v);
-      }
-    }
     bool items_created = false;
-    if(nm_vertices.empty() && warn)
+    if(nm_points.empty() && warn)
     {
       delete points;
       CGAL::Three::Three::information(tr("There is no non-manifold vertex in this soup."));
@@ -281,7 +264,7 @@ void Polyhedron_demo_orient_soup_plugin::createPointsAndPolyline(std::vector<std
     {
       points = new Scene_points_with_normal_item();
       items_created = true;
-      for(std::size_t id : nm_vertices)
+      for(std::size_t id : nm_points)
       {
         points->point_set()->insert(item->points()[id]);
       }
diff --git a/Polyhedron/demo/Polyhedron/Scene_polygon_soup_item.cpp b/Polyhedron/demo/Polyhedron/Scene_polygon_soup_item.cpp
index 9e5037a1105..116ff34f6b1 100644
--- a/Polyhedron/demo/Polyhedron/Scene_polygon_soup_item.cpp
+++ b/Polyhedron/demo/Polyhedron/Scene_polygon_soup_item.cpp
@@ -424,14 +424,45 @@ Scene_polygon_soup_item::orient(std::vector<std::size_t>& non_manifold_vertices)
 {
   struct Visitor : public CGAL::Polygon_mesh_processing::Default_orientation_visitor
   {
-    std::vector<std::size_t>& nm_vertices;
+    const Polygon_soup::Polygons& polygons;
+    std::set<std::size_t>& nm_vertices;
+    std::set< std::pair<std::size_t, std::size_t> > nm_edges;
 
-    Visitor(std::vector<std::size_t>& nm_vertices)
-      :nm_vertices(nm_vertices){}
+    Visitor(const Polygon_soup::Polygons& polygons,
+            std::set<std::size_t>& nm_vertices)
+      : polygons(polygons)
+      , nm_vertices(nm_vertices)
+    {}
 
-    void non_manifold_vertex(std::size_t v) final
+    void non_manifold_edge(std::size_t v1, std::size_t v2, std::size_t)
     {
-      nm_vertices.push_back(v);
+      nm_edges.insert(CGAL::make_sorted_pair(v1, v2));
+    }
+
+    void link_connected_polygons(std::size_t v, const std::vector<std::size_t>& polygons_in_current_cycle)
+    {
+      // check if the current components of polygon incident to the link of the vertex contains
+      // a non-manifold edge. If no, then it is a "pure" non-manifold vertex in this component
+      if (!nm_edges.empty())
+        for(std::size_t pid : polygons_in_current_cycle)
+        {
+          for (std::size_t i=0; i<polygons[pid].size(); ++i)
+          {
+            if (polygons[pid][i]==v)
+            {
+              std::size_t vn1 = i==0?(polygons[pid].size()-1):(i-1);
+              std::size_t vn2 = i+1;
+              if (vn2==polygons[pid].size()) vn2=0;
+
+              if (nm_edges.count(CGAL::make_sorted_pair(polygons[pid][i], polygons[pid][vn1]))==1) return;
+              if (nm_edges.count(CGAL::make_sorted_pair(polygons[pid][i], polygons[pid][vn2]))==1) return;
+              break;
+            }
+            else
+              CGAL_assertion(i!=polygons[pid].size()-1); // the vertex is incident to the polygon
+          }
+        }
+      nm_vertices.insert(v);
     }
   };
   if(isEmpty() || d->oriented)
@@ -460,10 +491,12 @@ Scene_polygon_soup_item::orient(std::vector<std::size_t>& non_manifold_vertices)
     d->soup->polygons.swap(valid_polygons);
 
   bool res;
-  Visitor visitor(non_manifold_vertices);
+  std::set<std::size_t> nm_v_set;
+  Visitor visitor(valid_polygons, nm_v_set);
   QApplication::setOverrideCursor(Qt::WaitCursor);
   res =  CGAL::Polygon_mesh_processing::
     orient_polygon_soup(d->soup->points, d->soup->polygons, CGAL::parameters::visitor(visitor));
+  non_manifold_vertices.assign(nm_v_set.begin(), nm_v_set.end());
   QApplication::restoreOverrideCursor();
   return res;
 }