diff --git a/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_2.C b/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_2.C
index 72a0baa4d2d..5589a6d97c0 100644
--- a/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_2.C
+++ b/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_2.C
@@ -814,7 +814,8 @@ insert_segment(const Site_2& t, Vertex_handle vnear,
   if ( number_of_vertices() == 0 ) {
     Vertex_handle v0 = insert_first( t.source() );
     Vertex_handle v1 = insert_second( t.target() );
-    return insert_third(v0, v1);
+    Vertex_handle vs = insert_third(v0, v1);
+    return vs;
   } else {
     Vertex_handle v0, v1;
     if ( number_of_vertices() == 1 ) {
@@ -829,10 +830,11 @@ insert_segment(const Site_2& t, Vertex_handle vnear,
     }
 
     Storage_site_2 ss = create_storage_site(v0, v1);
-    return insert_segment2( t, ss, v0, false );
-    //    insert_point( t.source_site(), Vertex_handle() );
-    //    insert_point( t.target_site(), Vertex_handle() );
-  }
+    // we do not add vs in the vertex list; it is inserted inside
+    // the method insert_segment2
+    vs = insert_segment2( t, ss, v0, false );
+    return vs;
+  } // if ( number_of_vertices() == 0 ) {
 }
 
 
@@ -871,8 +873,8 @@ insert_segment2(const Site_2& t, const Storage_site_2& ss,
   } else {
     vnearest = vnear;
   }
-
 #endif
+
   CGAL_assertion( vnearest != Vertex_handle() );
   // MK: add here code that checks if the inserted segment has already
   // been inserted; MAYBE THIS IS NOT NEEDED; I ALREADY DO IT IN
@@ -971,12 +973,15 @@ insert_segment2(const Site_2& t, const Storage_site_2& ss,
   // segments are found
   if ( vcross.first ) {
     if ( t.is_segment() ) {
+      Intersections_tag itag;
       return insert_intersecting_segment(ss, t, vcross.second,
-					 Intersections_tag());
+					 itag);
       //      return vcross.second;
     }
   }
 
+  // no intersecting segment has been found; we insert the segment as
+  // usual...
   Vertex_handle v = create_vertex(ss);
 
   retriangulate_conflict_region(v, l, fm);
diff --git a/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_2.h b/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_2.h
index b99b9922bb6..d73b157e38c 100644
--- a/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_2.h
+++ b/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_2.h
@@ -919,10 +919,22 @@ protected:
   //  Vertex_handle  insert_third(const Point_2& p0, const Point_2& p1);
   Vertex_handle  insert_third(Vertex_handle v0, Vertex_handle v1);
 
+  Vertex_handle insert_point(const Point_2& p, Vertex_handle vnear);
+  Vertex_handle insert_point(const Storage_site_2& ss,
+			     const Site_2& t, Vertex_handle vnear);
+
+  Vertex_handle insert_segment(const Site_2& t, Vertex_handle vnear,
+			       bool insert_endpoints);
+
+  Vertex_handle insert_segment2(const Site_2& t,
+				const Storage_site_2& ss,
+				Vertex_handle vnear, bool insert_endpoints);
+
   template<class ITag>
   Vertex_handle  insert_intersecting_segment(const Storage_site_2& ss,
 					     const Site_2& t,
-					     Vertex_handle v, ITag tag)
+					     Vertex_handle v,
+					     ITag tag)
   {
     return insert_intersecting_segment_with_tag(ss, t, v, tag);
   }
@@ -937,15 +949,7 @@ protected:
 				       const Site_2& t,
 				       Vertex_handle v, Tag_true);
 
-  Vertex_handle insert_point(const Point_2& p, Vertex_handle vnear);
-  Vertex_handle insert_point(const Storage_site_2& ss,
-			     const Site_2& t, Vertex_handle vnear);
 
-  Vertex_handle insert_segment(const Site_2& t, Vertex_handle vnear,
-			       bool insert_endpoints);
-  Vertex_handle insert_segment2(const Site_2& t,
-				const Storage_site_2& ss,
-				Vertex_handle vnear, bool insert_endpoints);
 
   // methods for insertion
   void initialize_conflict_region(const Face_handle& f, List& l);
diff --git a/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_hierarchy_2.h b/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_hierarchy_2.h
index 07de0d914ad..7374886dec2 100644
--- a/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_hierarchy_2.h
+++ b/Packages/Segment_Voronoi_diagram_2/include/CGAL/Segment_Voronoi_diagram_hierarchy_2.h
@@ -251,6 +251,8 @@ private:
 			Vertex_handle vnear[svd_hierarchy_2__maxlevel],
 			bool force_point) const; 
   int random_level();
+
+  size_type find_level(Vertex_handle v) const;
 };
 
 template<class Gt, class STag, class PC, class DS, class LTag>
@@ -605,6 +607,16 @@ random_level()
   return l;
 }
 
+template<class Gt, class STag, class PC, class DS, class LTag>
+inline typename
+Segment_Voronoi_diagram_hierarchy_2<Gt,STag,PC,DS,LTag>::size_type
+Segment_Voronoi_diagram_hierarchy_2<Gt,STag,PC,DS,LTag>::
+find_level(Vertex_handle v) const
+{
+  size_type level = 0;
+  while ( v->up() != Vertex_handle() ) { level++; }
+  return level;
+}
 
 template<class Gt, class STag, class PC, class DS, class LTag>
 bool