diff --git a/Constrained_triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h b/Constrained_triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h
index c6c06f1e904..a43f3f3120f 100644
--- a/Constrained_triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h
+++ b/Constrained_triangulation_3/include/CGAL/Conforming_Delaunay_triangulation_3.h
@@ -930,6 +930,7 @@ protected:
 
   Compare_vertex_handle comp = {this};
   Constraint_hierarchy constraint_hierarchy = {comp};
+  static_assert(CGAL::is_nothrow_movable_v<Constraint_hierarchy>);
   Bbox_3 bbox{};
   double segment_vertex_epsilon = 1e-8;
   std::optional<double> max_bbox_edge_length;
diff --git a/Constrained_triangulation_3/test/Constrained_triangulation_3/test_constrained_Delaunay_triangulation_3.cpp b/Constrained_triangulation_3/test/Constrained_triangulation_3/test_constrained_Delaunay_triangulation_3.cpp
index 8fa77fdda72..0c41947c04d 100644
--- a/Constrained_triangulation_3/test/Constrained_triangulation_3/test_constrained_Delaunay_triangulation_3.cpp
+++ b/Constrained_triangulation_3/test/Constrained_triangulation_3/test_constrained_Delaunay_triangulation_3.cpp
@@ -8,6 +8,8 @@
 using K = CGAL::Exact_predicates_inexact_constructions_kernel;
 using CDT = CGAL::Conforming_constrained_Delaunay_triangulation_3<K>;
 
+static_assert(CGAL::is_nothrow_movable_v<CDT>);
+
 int main(int argc, char* argv[])
 {
   CGAL::Surface_mesh<K::Point_3> mesh;
diff --git a/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h b/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h
index 5f159044b65..2ad95d310ad 100644
--- a/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h
+++ b/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h
@@ -73,6 +73,7 @@ public:
   using Self = Constrained_triangulation_plus_2<Tr_>;
   using Base = Tr_;
   using Constraint_hierarchy = Ctp_2_hierarchy_type<Tr_>;
+  static_assert(CGAL::is_nothrow_movable_v<Constraint_hierarchy>);
 protected:
   const auto& hierarchy() const { return static_cast<const Constraint_hierarchy&>(*this); }
   auto& hierarchy() { return static_cast<Constraint_hierarchy&>(*this); }
diff --git a/Triangulation_2/include/CGAL/Triangulation_2/internal/Polyline_constraint_hierarchy_2.h b/Triangulation_2/include/CGAL/Triangulation_2/internal/Polyline_constraint_hierarchy_2.h
index cd41a903ff8..825c4c462ad 100644
--- a/Triangulation_2/include/CGAL/Triangulation_2/internal/Polyline_constraint_hierarchy_2.h
+++ b/Triangulation_2/include/CGAL/Triangulation_2/internal/Polyline_constraint_hierarchy_2.h
@@ -526,10 +526,10 @@ private:
 #endif
     {}
 
-    Compare               comp;
-    Sc_to_c_map           sc_to_c_map;
-    std::queue<size_type> free_ids;
-    Constraints_set       constraints_set;
+    Compare comp;
+    Sc_to_c_map sc_to_c_map;
+    std::stack<size_type, std::vector<size_type>> free_ids;
+    Constraints_set constraints_set;
   } priv;
 public:
   Polyline_constraint_hierarchy_2(const Compare& comp) : priv(comp)  {}
@@ -650,7 +650,7 @@ private:
     if(priv.free_ids.empty()) {
       id = priv.constraints_set.size();
     } else {
-      id = priv.free_ids.front();
+      id = priv.free_ids.top();
       priv.free_ids.pop();
     }
     Constraint_id cid{new Vertex_list_with_info{this}, id};