diff --git a/Polyline_simplification_2/package_info/Polyline_simplification_2/dependencies b/Polyline_simplification_2/package_info/Polyline_simplification_2/dependencies
index ba5bb8645f3..46b60927a44 100644
--- a/Polyline_simplification_2/package_info/Polyline_simplification_2/dependencies
+++ b/Polyline_simplification_2/package_info/Polyline_simplification_2/dependencies
@@ -1,5 +1,6 @@
 Algebraic_foundations
 Arithmetic_kernel
+BGL
 Cartesian_kernel
 Circulator
 Distance_2
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 821d59c2d42..7f8eeddc547 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 +293,30 @@ std::size_t insert_constraints(PointIterator points_first, PointIterator points_
                                IndicesIterator indices_first, IndicesIterator indices_last);
 
 
+/*!
+splits into constraints the graph of subconstraints.
+
+Consider the graph `g={V,E}` where `V` is the set of vertices of the
+triangulation and `E` is the set of all subconstraints of all
+constraints of the triangulation.
+
+This function splits into polylines the graph `g` at vertices of
+degree greater than 2 and at vertices for which
+`is_terminal(v)==true`.
+
+Each computed polyline is stored as a constraint of the triangulation.
+
+\warning all existing constraints will be discarded.
+
+\param is_terminal An optional function returning `true` if the vertex
+`v` of degree 2 is a polyline endpoint and `false` otherwise. If
+omitted, a function always returning `false` will be used, that is no
+degree 2 vertex will be considered as a polyline endpoint.
+
+\sa `split_graph_into_polylines()`
+*/
+void split_subconstraint_graph_into_constraints(const std::function<bool(Vertex_handle)>& is_terminal);
+
 /*!
 removes the constraint `cid`, without removing the points from the triangulation.
 */
diff --git a/Triangulation_2/doc/Triangulation_2/Triangulation_2.txt b/Triangulation_2/doc/Triangulation_2/Triangulation_2.txt
index 3c7e043b29a..90446677a4c 100644
--- a/Triangulation_2/doc/Triangulation_2/Triangulation_2.txt
+++ b/Triangulation_2/doc/Triangulation_2/Triangulation_2.txt
@@ -1124,6 +1124,22 @@ through the edge.
 
 \cgalExample{Triangulation_2/polylines_triangulation.cpp}
 
+\subsection Triangulation_2ExampleBuildingFromSoup Example: Building a Triangulated Arrangement of Polylines from a Segment Soup
+
+The `Constrained_triangulation_plus_2` structure is initialized by a
+set of polylines. As users may only be able to provide a disconnected
+segment soup as input, a member function \link
+Constrained_triangulation_plus_2::split_subconstraint_graph_into_constraints()
+`split_subconstraint_graph_into_constraints()` \endlink is provided:
+this function identifies the polylines by connecting segments until a
+vertex whose degree is different from 2 is reached.
+
+The following code shows how a "blind" insertion of disconnected
+segments can be processed into a set of well-defined polyline
+constraints in a `Constrained_triangulation_plus_2`:
+
+\cgalExample{Triangulation_2/segment_soup_to_polylines.cpp}
+
 \section Section_2D_Triangulations_Hierarchy The Triangulation Hierarchy
 
 The class `Triangulation_hierarchy_2<Tr>`
diff --git a/Triangulation_2/doc/Triangulation_2/dependencies b/Triangulation_2/doc/Triangulation_2/dependencies
index 84dba85f07d..22ae69dcfef 100644
--- a/Triangulation_2/doc/Triangulation_2/dependencies
+++ b/Triangulation_2/doc/Triangulation_2/dependencies
@@ -4,6 +4,7 @@ STL_Extension
 Algebraic_foundations
 Circulator
 Stream_support
+BGL
 TDS_2
 Triangulation_3
 Spatial_sorting
diff --git a/Triangulation_2/doc/Triangulation_2/examples.txt b/Triangulation_2/doc/Triangulation_2/examples.txt
index 45128d71309..7061593de13 100644
--- a/Triangulation_2/doc/Triangulation_2/examples.txt
+++ b/Triangulation_2/doc/Triangulation_2/examples.txt
@@ -19,5 +19,6 @@
 \example Triangulation_2/voronoi.cpp
 \example Triangulation_2/copy_triangulation_2.cpp
 \example Triangulation_2/polylines_triangulation.cpp
+\example Triangulation_2/segment_soup_to_polylines.cpp
 \example Triangulation_2/draw_triangulation_2.cpp
 */
diff --git a/Triangulation_2/examples/Triangulation_2/segment_soup_to_polylines.cpp b/Triangulation_2/examples/Triangulation_2/segment_soup_to_polylines.cpp
new file mode 100644
index 00000000000..5eadd29090f
--- /dev/null
+++ b/Triangulation_2/examples/Triangulation_2/segment_soup_to_polylines.cpp
@@ -0,0 +1,51 @@
+#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
+#include <CGAL/Constrained_Delaunay_triangulation_2.h>
+#include <CGAL/Constrained_triangulation_plus_2.h>
+
+typedef CGAL::Exact_predicates_exact_constructions_kernel                 K;
+typedef CGAL::Constrained_Delaunay_triangulation_2<K>                     CDT;
+typedef CGAL::Constrained_triangulation_plus_2<CDT>                       CDTP;
+
+typedef CDTP::Point                                                       Point;
+typedef CDTP::Constraint_id                                               Cid;
+typedef CDTP::Vertex_handle                                               Vertex_handle;
+
+int main()
+{
+  CDTP cdtp;
+
+  // First polyline
+  cdtp.insert_constraint(Point(0,0), Point(1,1));
+  cdtp.insert_constraint(Point(1,1), Point(2,2));
+  cdtp.insert_constraint(Point(2,2), Point(3,3));
+
+  // Second polyline that cuts the first one in 2
+  cdtp.insert_constraint(Point(1,1), Point(1,2));
+  cdtp.insert_constraint(Point(1,2), Point(1,3));
+
+  // Third polyline
+  cdtp.insert_constraint(Point(10,10), Point(20,20));
+  cdtp.insert_constraint(Point(20,20), Point(30,30));
+
+  // Fourth polyline
+  cdtp.insert_constraint(Point(100,100), Point(200,200));
+
+  // Segment soup of 8 segments as input
+  std::cout << "Input CDT+ has " << cdtp.number_of_constraints() << " constraints/subconstraints" << std::endl;
+
+  std::cout << "Splitting subconstraints graph into constraints" << std::endl;
+  cdtp.split_subconstraint_graph_into_constraints();
+
+  // 5 polylines as output
+  std::cout << "Output CDT+ has " << cdtp.number_of_constraints() << " constraints:" << std::endl;
+
+  for (CDTP::Constraint_id cid : cdtp.constraints())
+  {
+    std::cout << " *";
+    for (CDTP::Vertex_handle vh : cdtp.vertices_in_constraint(cid))
+      std::cout << " (" << vh->point() << ")";
+    std::cout << std::endl;
+  }
+
+  return EXIT_SUCCESS;
+}
diff --git a/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h b/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h
index 51d291d1454..0b19a9a5698 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..77e212b9f04
--- /dev/null
+++ b/Triangulation_2/include/CGAL/Triangulation_2/internal/CTP2_subconstraint_graph.h
@@ -0,0 +1,126 @@
+// 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>
+
+#include <CGAL/property_map.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&)
+  {
+    return ed.first;
+  }
+
+  friend vertex_descriptor target (edge_descriptor ed, const CTP2_subconstraint_graph&)
+  {
+    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