diff --git a/Triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h b/Triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h
index dc7d7f5aeeb..50a88aaeab9 100644
--- a/Triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h
+++ b/Triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h
@@ -42,12 +42,19 @@ namespace CGAL {
 
 enum class CDT_3_vertex_type { FREE, CORNER, STEINER_ON_EDGE, STEINER_IN_FACE };
 
+enum class CDT_3_vertex_marker {
+  CLEAR = 0,
+  REGION_BORDER,
+  REGION_INSIDE,
+  nb_of_markers
+};
+
 using CDT_3_face_index = int; // must be signed
 
 template <typename Gt, typename Vb = Triangulation_vertex_base_3<Gt> >
 class Conforming_Delaunay_triangulation_vertex_base_3 : public Base_with_time_stamp<Vb> {
   CDT_3_vertex_type m_vertex_type = CDT_3_vertex_type::FREE;
-  unsigned int mark = 0;
+  std::bitset<static_cast<int>(CDT_3_vertex_marker::nb_of_markers)> mark;
   union U {
     struct On_edge {
       int nb_of_incident_constraints = 0;
@@ -83,16 +90,20 @@ public:
     return u.on_edge.nb_of_incident_constraints;
   }
 
-  void mark_vertex() {
-    mark = 1;
+  void set_mark(CDT_3_vertex_marker marker) {
+    mark.set(static_cast<unsigned int>(marker));
   }
 
-  void unmark_vertex() {
-    mark = 0;
+  void clear_marks() {
+    mark.reset();
   }
 
-  bool is_marked() const {
-    return mark == 1;
+  void clear_mark(CDT_3_vertex_marker marker) {
+    mark.reset(static_cast<unsigned int>(marker));
+  }
+
+  bool is_marked(CDT_3_vertex_marker marker) const {
+    return mark.test(static_cast<unsigned int>(marker));
   }
 
   template<typename Triangulation>
diff --git a/Triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h b/Triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h
index ded146abb5d..f541a5fc81a 100644
--- a/Triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h
+++ b/Triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_3.h
@@ -218,6 +218,8 @@ public:
   using Locate_type = typename T_3::Locate_type;
   using Geom_traits = typename T_3::Geom_traits;
 
+  using Vertex_marker = CDT_3_vertex_marker;
+
   using Face_index = CDT_3_face_index;
 
   static std::string io_signature() {
@@ -1026,8 +1028,8 @@ private:
   {
     const auto vc = cell->vertex(index_vd);
     const auto vd = cell->vertex(index_vc);
-    if(!vd->is_Steiner_vertex_in_face() && vd->is_marked()) return 0; // vertex marked of the border
-    if(!vc->is_Steiner_vertex_in_face() && vc->is_marked()) return 0; // vertex marked of the border
+    if(vc->is_marked(Vertex_marker::REGION_BORDER)) return 0; // vertex marked of the border
+    if(vd->is_marked(Vertex_marker::REGION_BORDER)) return 0; // vertex marked of the border
     const auto pc = this->point(vc);
     const auto pd = this->point(vd);
     const typename Geom_traits::Segment_3 seg{pc, pd};
@@ -1200,10 +1202,10 @@ private:
 #endif // CGAL_CDT_3_CAN_USE_CXX20_FORMAT
           auto [cached_value_it, not_visited] = new_edge(v0, v1, false);
           if(!not_visited) return value_returned(cached_value_it->second);
-          int v0v1_intersects_region = ((v0->is_Steiner_vertex_in_face() && v0->face_index() == face_index) ||
-                                        (v1->is_Steiner_vertex_in_face() && v1->face_index() == face_index))
-                                           ? expected
-                                           : does_edge_intersect_region(cell, index_v0, index_v1, cdt_2, fh_region);
+          int v0v1_intersects_region =
+              (v0->is_marked(Vertex_marker::REGION_INSIDE) || v1->is_marked(Vertex_marker::REGION_INSIDE))
+                  ? expected
+                  : does_edge_intersect_region(cell, index_v0, index_v1, cdt_2, fh_region);
           if(v0v1_intersects_region != 0) {
             if(v0v1_intersects_region != expected) {
               throw PLC_error{"PLC error: v0v1_intersects_region != expected" ,
@@ -1306,11 +1308,11 @@ private:
     }
 #endif // CGAL_CDT_3_CAN_USE_CXX20_FORMAT
     for(auto v: region_border_vertices) {
-      v->mark_vertex();
+      v->set_mark(Vertex_marker::REGION_BORDER);
     }
     const auto found_edge_opt = search_first_intersection(face_index, cdt_2, fh_region, border_edges);
     for(auto v: region_border_vertices) {
-      v->unmark_vertex();
+      v->clear_mark(Vertex_marker::REGION_BORDER);
     }
 
     [[maybe_unused]] auto try_flip_region_size_4 = [&] {
@@ -1448,6 +1450,25 @@ private:
     }
     CGAL_assertion(found_edge_opt != std::nullopt);
 
+    for(auto v : region_border_vertices) {
+      v->set_mark(Vertex_marker::REGION_BORDER);
+    }
+    for(auto v : region_vertices) {
+      if(v->is_marked(Vertex_marker::REGION_BORDER))
+        continue;
+      v->set_mark(Vertex_marker::REGION_INSIDE);
+    }
+    struct Region_vertices_marker_scope_guard
+    {
+      std::function<void()> unmark;
+      ~Region_vertices_marker_scope_guard() { unmark(); }
+    } guard{[&] {
+      for(auto v : region_vertices) {
+        v->clear_mark(Vertex_marker::REGION_BORDER);
+        v->clear_mark(Vertex_marker::REGION_INSIDE);
+      }
+    }};
+
     const auto first_intersecting_edge = *found_edge_opt;
     const auto [intersecting_edges, original_intersecting_cells, original_vertices_of_upper_cavity,
                 original_vertices_of_lower_cavity, original_facets_of_upper_cavity, original_facets_of_lower_cavity] =