diff --git a/Triangulation_2/doc/Triangulation_2/CGAL/Constrained_triangulation_plus_2.h b/Triangulation_2/doc/Triangulation_2/CGAL/Constrained_triangulation_plus_2.h
index 4fb60c08fbc..cef1af9c55f 100644
--- a/Triangulation_2/doc/Triangulation_2/CGAL/Constrained_triangulation_plus_2.h
+++ b/Triangulation_2/doc/Triangulation_2/CGAL/Constrained_triangulation_plus_2.h
@@ -293,6 +293,13 @@ std::size_t insert_constraints(PointIterator points_first, PointIterator points_
                                IndicesIterator indices_first, IndicesIterator indices_last);
 
 
+/*! 
+splits into constraints the subconstraint graph g at vertices of degree greater than 2.
+
+\warning all existing constraints will be discarded.
+*/ 
+void split_subconstraint_graph_into_constraints();
+  
 /*! 
 removes the constraint `cid`, without removing the points from the triangulation.
 */ 
diff --git a/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h b/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h
index dad024bd902..676952750b9 100644
--- a/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h
+++ b/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h
@@ -21,6 +21,7 @@
 #include <CGAL/triangulation_assertions.h>
 #include <CGAL/Polygon_2.h>
 #include <CGAL/Triangulation_2/internal/Polyline_constraint_hierarchy_2.h>
+#include <CGAL/Triangulation_2/internal/CTP2_subconstraint_graph.h>
 #include <boost/tuple/tuple.hpp>
 #include <boost/type_traits/is_same.hpp>
 
@@ -315,6 +316,22 @@ public:
     return n;
   }
   */
+
+  void split_subconstraint_graph_into_constraints(const std::function<bool(Vertex_handle)>& is_terminal
+                                                  = std::function<bool(Vertex_handle)>())
+  {
+    internal::CTP2_graph_visitor<Self> visitor(*this);
+    if (is_terminal)
+      CGAL::split_graph_into_polylines (internal::CTP2_subconstraint_graph<Self>(*this), visitor,
+                                        [&is_terminal](Vertex_handle vh,
+                                                       const internal::CTP2_subconstraint_graph<Self>&) -> bool
+                                        {
+                                          return is_terminal(vh);
+                                        });
+    else
+      CGAL::split_graph_into_polylines (internal::CTP2_subconstraint_graph<Self>(*this), visitor);
+  }
+
   Vertex_handle push_back(const Point& p)
   {
     return insert(p);
diff --git a/Triangulation_2/include/CGAL/Triangulation_2/internal/CTP2_subconstraint_graph.h b/Triangulation_2/include/CGAL/Triangulation_2/internal/CTP2_subconstraint_graph.h
new file mode 100644
index 00000000000..9e1346d3d2c
--- /dev/null
+++ b/Triangulation_2/include/CGAL/Triangulation_2/internal/CTP2_subconstraint_graph.h
@@ -0,0 +1,124 @@
+// Copyright (c) 2020  GeometryFactory (France).  All rights reserved.
+//
+// This file is part of CGAL (www.cgal.org)
+//
+// $URL$
+// $Id$
+// SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial
+// 
+//
+// Author(s)     : Simon Giraudot
+
+#ifndef CGAL_CTP2_SUBCONSTRAINT_GRAPH_H
+#define CGAL_CTP2_SUBCONSTRAINT_GRAPH_H
+
+#include <CGAL/license/Triangulation_2.h>
+
+#include <CGAL/boost/graph/internal/graph_traits_2D_triangulation_helper.h>
+#include <CGAL/boost/graph/split_graph_into_polylines.h>
+
+namespace CGAL
+{
+
+namespace internal
+{
+
+template <typename CTP2>
+class CTP2_subconstraint_graph
+{
+  CTP2& ctp2;
+public:
+
+  typedef typename CTP2::Vertex_handle vertex_descriptor;
+  typedef typename CTP2::Subconstraint edge_descriptor;
+  typedef boost::undirected_tag directed_category;
+  typedef boost::disallow_parallel_edge_tag edge_parallel_category;
+  struct CTP2_graph_traversal_category :
+    public virtual boost::bidirectional_graph_tag,
+    public virtual boost::adjacency_graph_tag,
+    public virtual boost::edge_list_graph_tag,
+    public virtual boost::vertex_list_graph_tag
+  { };
+  typedef CTP2_graph_traversal_category traversal_category;
+  typedef internal::Dereference_to_handle_enforcer<
+            CTP2,
+            typename CTP2::Finite_vertices_iterator,
+            vertex_descriptor>                                                vertex_iterator;
+
+  typedef typename CTP2::Subconstraint_iterator::value_type Subconstr_it_v_t;
+  typedef First_of_pair_property_map<Subconstr_it_v_t> Subconstr_map;
+  typedef Property_map_to_unary_function<Subconstr_map> Subconstr_uf;
+  typedef boost::transform_iterator<Subconstr_uf, typename CTP2::Subconstraint_iterator> edge_iterator;
+
+  CTP2_subconstraint_graph (CTP2& ctp2) : ctp2(ctp2) { }
+  
+  friend Iterator_range<vertex_iterator> vertices (const CTP2_subconstraint_graph& g)
+  {
+    return make_range (vertex_iterator(g.ctp2.finite_vertices_begin()),
+                       vertex_iterator(g.ctp2.finite_vertices_end()));
+  }
+  
+  friend Iterator_range<edge_iterator> edges (const CTP2_subconstraint_graph& g)
+  {
+    return make_range (boost::make_transform_iterator(g.ctp2.subconstraints_begin(), Subconstr_uf(Subconstr_map())),
+                       boost::make_transform_iterator(g.ctp2.subconstraints_end(), Subconstr_uf(Subconstr_map())));
+  }
+  
+  friend vertex_descriptor source (edge_descriptor ed, const CTP2_subconstraint_graph& g)
+  {
+    return ed.first;
+  }
+  
+  friend vertex_descriptor target (edge_descriptor ed, const CTP2_subconstraint_graph& g)
+  {
+    return ed.second;
+  }
+};
+
+
+template <typename CTP2>
+class CTP2_graph_visitor
+{
+private:
+  CTP2& ctp2;
+  std::vector<typename CTP2::Constraint_id> to_remove;
+  typename CTP2::Constraint_id current;
+  typename CTP2::Vertex_handle latest_vertex;
+
+public:
+
+  CTP2_graph_visitor (CTP2& ctp2) : ctp2 (ctp2) { }
+  
+  void start_new_polyline()
+  {
+    latest_vertex = typename CTP2::Vertex_handle();
+    current = typename CTP2::Constraint_id();
+  }
+  
+  void add_node (typename CTP2::Vertex_handle vh)
+  {
+    if (latest_vertex != typename CTP2::Vertex_handle())
+    {
+      to_remove.push_back (ctp2.context(latest_vertex, vh).id());
+      typename CTP2::Constraint_id cid = ctp2.insert_constraint(latest_vertex, vh);
+      if (current == typename CTP2::Constraint_id())
+        current = cid;
+      else
+        current = ctp2.concatenate (current, cid);
+    }
+    latest_vertex = vh;
+  }
+  
+  void end_polyline()
+  {
+    for (typename CTP2::Constraint_id id : to_remove)
+      ctp2.remove_constraint(id);
+    to_remove.clear();
+  }
+};
+
+} // namespace internal
+
+} // namespace CGAL
+
+#endif // CGAL_CTP2_SUBCONSTRAINT_GRAPH_H