diff --git a/Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_insertion_ss_visitor.h b/Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_insertion_ss_visitor.h
index 6782f6c02b6..a9ab34bd676 100644
--- a/Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_insertion_ss_visitor.h
+++ b/Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_insertion_ss_visitor.h
@@ -67,6 +67,36 @@ private:
   X_monotone_curve_2 sub_cv1;         // Auxiliary variables
   X_monotone_curve_2 sub_cv2;         // (used for splitting curves).
 
+
+  // update halfedge pointing to events, case with overlaps
+  template <class Subcurve_>
+  void update_incident_halfedge_after_split(Subcurve_* sc,
+                                            Halfedge_handle he,
+                                            Halfedge_handle new_he,
+                                            Tag_true)
+  {
+    std::vector<Subcurve*> leaves;
+    sc->all_leaves( std::back_inserter(leaves) );
+    for(Subcurve* ssc : leaves)
+    {
+      Event* last_event_on_ssc = ssc->last_event();
+      if (last_event_on_ssc->halfedge_handle() == he)
+        last_event_on_ssc->set_halfedge_handle(new_he->next());
+    }
+  }
+
+  // update halfedge pointing to events, case without overlaps
+  template <class Subcurve_>
+  void update_incident_halfedge_after_split(Subcurve_* sc,
+                                            Halfedge_handle he,
+                                            Halfedge_handle new_he,
+                                            Tag_false)
+  {
+    Event* last_event_on_sc = sc->last_event();
+    if (last_event_on_sc->halfedge_handle() == he)
+      last_event_on_sc->set_halfedge_handle(new_he->next());
+  }
+
 public:
   /*! A notification invoked when a new subcurve is created. */
   void add_subcurve(const X_monotone_curve_2& cv, Subcurve* sc);
@@ -132,9 +162,8 @@ Arr_insertion_ss_visitor<Hlpr, Vis>::split_edge(Halfedge_handle he, Subcurve* sc
   Halfedge_handle new_he =
     this->m_arr_access.split_edge_ex(he, pt.base(),
                                      sub_cv1.base(), sub_cv2.base());
-  Event* last_event_on_sc = sc->last_event();
-  if (last_event_on_sc->halfedge_handle() == he)
-    last_event_on_sc->set_halfedge_handle(new_he->next());
+  // update the halfedge incident to events on the left of the split
+  update_incident_halfedge_after_split(sc, he, new_he, typename Subcurve::Handle_overlaps());
 
   return new_he;
 }
diff --git a/Surface_sweep_2/include/CGAL/Surface_sweep_2/Default_subcurve.h b/Surface_sweep_2/include/CGAL/Surface_sweep_2/Default_subcurve.h
index 1112d578cd4..dd8215e67a1 100644
--- a/Surface_sweep_2/include/CGAL/Surface_sweep_2/Default_subcurve.h
+++ b/Surface_sweep_2/include/CGAL/Surface_sweep_2/Default_subcurve.h
@@ -70,6 +70,7 @@ public:
   typedef GeometryTraits_2                              Geometry_traits_2;
   typedef Subcurve_                                     Subcurve;
   typedef Event_                                        Event;
+  typedef Tag_true                                      Handle_overlaps;
 
 private:
   typedef Geometry_traits_2                             Gt2;
diff --git a/Surface_sweep_2/include/CGAL/Surface_sweep_2/No_overlap_subcurve.h b/Surface_sweep_2/include/CGAL/Surface_sweep_2/No_overlap_subcurve.h
index bef25a3d9b9..1fd3601f077 100644
--- a/Surface_sweep_2/include/CGAL/Surface_sweep_2/No_overlap_subcurve.h
+++ b/Surface_sweep_2/include/CGAL/Surface_sweep_2/No_overlap_subcurve.h
@@ -63,6 +63,7 @@ public:
   typedef Subcurve_                                     Subcurve;
   typedef Event_                                        Event;
   typedef Allocator_                                    Allocator;
+  typedef Tag_false                                     Handle_overlaps;
 
 private:
   typedef Geometry_traits_2                             Gt2;