diff --git a/BGL/include/CGAL/BGL/Free_iterator_from_circulator.h b/BGL/include/CGAL/BGL/Free_iterator_from_circulator.h
deleted file mode 100644
index 13521c0815a..00000000000
--- a/BGL/include/CGAL/BGL/Free_iterator_from_circulator.h
+++ /dev/null
@@ -1,187 +0,0 @@
-// Copyright (c) 2012 GeometryFactory (France). All rights reserved.
-// All rights reserved.
-//
-// Licensees holding a valid commercial license may use this file in
-// accordance with the commercial license agreement provided with the software.
-//
-// This file is part of CGAL (www.cgal.org); you can redistribute it and/or
-// modify it under the terms of the GNU Lesser General Public License as
-// published by the Free Software Foundation; version 2.1 of the License.
-// See the file LICENSE.LGPL distributed with CGAL.
-//
-// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
-// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
-//
-// $URL$
-// $Id$
-// 
-//
-// Author(s)     : Philipp Moeller
-
-#ifndef CGAL_FREE_ITERATOR_FROM_CIRCULATOR_H
-#define CGAL_FREE_ITERATOR_FROM_CIRCULATOR_H
-
-
-#include <iostream>
-#include <cstddef>
-
-#include <CGAL/circulator.h>
-
-#include <boost/operators.hpp>
-#include <boost/concept_check.hpp>
-
-#include <boost/mpl/if.hpp>
-#include <boost/utility/enable_if.hpp>
-
-namespace CGAL {
-
-// adapted from circulator.h, does not support
-// random_access_circulators and returns the underlying circulator
-// instead of dereferencing it
-template <class C, bool Prevent_deref = true>
-class Free_iterator_from_circulator {
-private:
-  // The m_anchor is normalized to be a minimal circulator.
-  C         m_anchor;
-  C         current;
-  int       m_winding;
-
-  typedef  std::iterator_traits<C>                       I_traits;
-  typedef  typename  I_traits::iterator_category         I_Iter_cat;
-  typedef  I_Iterator_from_circulator_traits<I_Iter_cat> I__traits;
-
-public:
-  typedef C                                               Circulator;
-  typedef Free_iterator_from_circulator<C, Prevent_deref> Self;
-
-  typedef typename I__traits::iterator_category iterator_category;
-
-  typedef typename 
-  boost::mpl::if_c<  Prevent_deref
-                   , C
-                   , typename C::value_type
-                  >::type             value_type;
-
-  typedef typename C::difference_type difference_type;
-  typedef typename 
-  boost::mpl::if_c<  Prevent_deref
-                   , C&
-                   , typename C::reference
-                  >::type             reference;
-  typedef typename 
-  boost::mpl::if_c<  Prevent_deref
-                   , C*
-                   , typename C::reference
-                  >::type             pointer;
-
-  Free_iterator_from_circulator(){}
-
-  Free_iterator_from_circulator(const C circ, int n)
-    : m_anchor(circ), current(circ), m_winding(n) {}
-
-  bool operator==( const Self& i) const {
-    CGAL_assertion( m_anchor == i.m_anchor);  // same anchor?
-    return ( current == i.current) && ( m_winding == i.m_winding);
-  }
-  
-  bool operator!=( const Self& i) const {
-    return !(*this == i);
-  }
-
-
-// we cannot enable_if on operator* and operator-> because they do not
-// depend on template parameters directly and default template
-// arguments for functions are C++11. we redirect on helper member
-// templates as a work-around.
-private:
-  template <bool Prevent_deref_>
-  typename boost::enable_if_c<Prevent_deref_, reference>::type
-  indirection() const {
-    CGAL_assertion( current  != NULL);
-    return const_cast<Self*>(this)->current;
-  }
-  template <bool Prevent_deref_>
-  typename boost::disable_if_c<Prevent_deref_, reference>::type
-  indirection() const {
-    CGAL_assertion( current  != NULL);
-    return *current;
-  }
-public:
-  reference operator*() const {
-    return indirection<Prevent_deref>();
-  }
-  
-private:
-  template <bool Prevent_deref_>
-  typename boost::disable_if_c<Prevent_deref_, pointer>::type
-  structure_dereference() {
-    CGAL_assertion( current  != NULL);
-    return &(*current);
-  }
-  template <bool Prevent_deref_>
-  typename boost::enable_if_c<Prevent_deref_, pointer>::type
-  structure_dereference() {
-    CGAL_assertion( current  != NULL);
-    return &current;
-  }
-public:
-  pointer operator->() const {
-    return structure_dereference<Prevent_deref>();
-  }
-
-  Self& operator++() {
-    CGAL_assertion( current  != NULL);
-    ++current;
-    if ( current == m_anchor)
-      ++m_winding;
-    return *this;
-  }
-  Self  operator++(int) {
-    Self tmp = *this;
-    ++*this;
-    return tmp;
-  }
-  Self& operator--() {
-    CGAL_assertion( current != NULL);
-    if ( current == m_anchor)
-      --m_winding;
-    --current;
-    return *this;
-  }
-  Self  operator--(int) {
-    Self tmp = *this;
-    --*this;
-    return tmp;
-  }
-  Self& operator+=( difference_type n) {
-    CGAL_assertion( current != NULL);
-    if ( n < 0 && current == m_anchor)  // We are leaving the anchor.
-      --m_winding;
-    current += n;
-    if ( n > 0 && current == m_anchor)  // Back again at the anchor.
-      ++m_winding;
-    return *this;
-  }
-
-  bool operator<( const Self& i) const {
-    CGAL_assertion( m_anchor  != NULL);
-    CGAL_assertion( current != NULL);
-    CGAL_assertion( m_anchor  == i.m_anchor);
-    return (     (m_winding < i.m_winding)
-                 || (    (m_winding == i.m_winding)
-                         && (current - m_anchor) < (i.current - m_anchor)
-                   )
-      );
-  }
-  bool operator> ( const Self& i) const { return i < *this; }
-  bool operator<=( const Self& i) const { return !(i < *this); }
-  bool operator>=( const Self& i) const { return !(*this < i); }
-
-  const C*    anchor()             const { return m_anchor;}
-  int         winding()            const { return m_winding;}
-  Circulator  current_circulator() const { return current;}
-};
-
-} // CGAL
-
-#endif /* CGAL_FREE_ITERATOR_FROM_CIRCULATOR_H */
diff --git a/BGL/include/CGAL/boost/graph/Euler_operations.h b/BGL/include/CGAL/boost/graph/Euler_operations.h
index d401fa8c406..f384a9c7323 100644
--- a/BGL/include/CGAL/boost/graph/Euler_operations.h
+++ b/BGL/include/CGAL/boost/graph/Euler_operations.h
@@ -26,7 +26,8 @@
 #include <boost/graph/graph_traits.hpp>
 
 #include <CGAL/assertions.h>
-#include <CGAL/BGL/Helper.h>
+#include <CGAL/boost/graph/helpers.h>
+#include <CGAL/boost/graph/internal/helpers.h>
 
 namespace CGAL {
 
@@ -51,10 +52,10 @@ join_face(typename boost::graph_traits<Graph>::halfedge_descriptor h,
 
   internal::remove_tip(gprev, g);
 
-  if(! internal::is_border(hop,g)){
+  if(! is_border(hop,g)){
     remove_face(f2, g);
   }
-  bool fnull = internal::is_border(h,g);
+  bool fnull = is_border(h,g);
 
   
   halfedge_descriptor hprev2 = hprev;
@@ -322,10 +323,10 @@ join_loop(typename boost::graph_traits<Graph>::halfedge_descriptor h1,
   typedef typename Traits::halfedge_descriptor             halfedge_descriptor;
   typedef typename Traits::face_descriptor                 face_descriptor;
 
-  CGAL_precondition( internal::is_border(h1,g) || face(h1, g) != face(h2, g));
-  if (! internal::is_border(h1,g))
+  CGAL_precondition( is_border(h1,g) || face(h1, g) != face(h2, g));
+  if (! is_border(h1,g))
     remove_face(face(h1, g), g);
-  if (! internal::is_border(h2,g))
+  if (! is_border(h2,g))
     remove_face(face(h2,g), g);
   halfedge_descriptor hi = h1;
   halfedge_descriptor gi = h2;
@@ -497,15 +498,15 @@ void remove_face(typename boost::graph_traits<Graph>::halfedge_descriptor h,
   typedef typename Traits::halfedge_descriptor           halfedge_descriptor;
   typedef typename Traits::face_descriptor               face_descriptor;
 
-  CGAL_precondition(! internal::is_border(h,g));
+  CGAL_precondition(! is_border(h,g));
   face_descriptor f = face(h, g);
 
   halfedge_descriptor end = h;
   do {
     internal::set_border(h,g);
     halfedge_descriptor nh = next(h, g);
-    bool h_border = internal::is_border(opposite(h, g),g);
-    bool nh_bborder = internal::is_border(opposite(nh, g),g);
+    bool h_border = is_border(opposite(h, g),g);
+    bool nh_bborder = is_border(opposite(nh, g),g);
 
     if(h_border && nh_bborder && next(opposite(nh, g), g) == opposite(h, g)) {
       remove_vertex(target(h, g), g);
@@ -528,7 +529,7 @@ void remove_face(typename boost::graph_traits<Graph>::halfedge_descriptor h,
     h = nh;
   } while(h != end);
   remove_face(f, g);
-  if(internal::is_border(opposite(h, g),g))
+  if(is_border(opposite(h, g),g))
     remove_edge(edge(h, g), g);
 }
 
@@ -547,13 +548,13 @@ void make_hole(typename boost::graph_traits<Graph>::halfedge_descriptor h,
   typedef typename Traits::face_descriptor               face_descriptor;
   typedef Halfedge_around_face_iterator<Graph>                   halfedge_around_face_iterator;
 
-  CGAL_precondition(! internal::is_border(h,g));
+  CGAL_precondition(! is_border(h,g));
   face_descriptor fd = face(h, g);
   halfedge_around_face_iterator hafib, hafie;
   for(boost::tie(hafib, hafie) = halfedges_around_face(h, g); 
       hafib != hafie; 
       ++hafib){
-    CGAL_assertion(! internal::is_border(opposite(*hafib,g),g));
+    CGAL_assertion(! is_border(opposite(*hafib,g),g));
     internal::set_border(*hafib, g);
   }
   remove_face(fd,g);  
@@ -718,8 +719,8 @@ add_face_to_border(typename boost::graph_traits<Graph>::halfedge_descriptor h1,
                    typename boost::graph_traits<Graph>::halfedge_descriptor h2,
                    Graph& g)
 {
-  CGAL_precondition(internal::is_border(h1,g) == true);
-  CGAL_precondition(internal::is_border(h2,g) == true);
+  CGAL_precondition(is_border(h1,g) == true);
+  CGAL_precondition(is_border(h2,g) == true);
   CGAL_precondition(h1 != h2);
   CGAL_precondition(next(h1, g) != h2);
 
@@ -800,10 +801,10 @@ collapse_edge(typename boost::graph_traits<Graph>::edge_descriptor v0v1,
   halfedge_descriptor pt = opposite(prev(pq, g), g);
   halfedge_descriptor qb = opposite(prev(qp, g), g);
   
-  bool lTopFaceExists         = ! internal::is_border(pq,g);
-  bool lBottomFaceExists      = ! internal::is_border(qp,g);
-  bool lTopLeftFaceExists     = lTopFaceExists    && ! internal::is_border(pt,g);
-  bool lBottomRightFaceExists = lBottomFaceExists && ! internal::is_border(qb,g);
+  bool lTopFaceExists         = ! is_border(pq,g);
+  bool lBottomFaceExists      = ! is_border(qp,g);
+  bool lTopLeftFaceExists     = lTopFaceExists    && ! is_border(pt,g);
+  bool lBottomRightFaceExists = lBottomFaceExists && ! is_border(qb,g);
 
   CGAL_precondition( !lTopFaceExists    || (lTopFaceExists    && ( degree(target(pt, g), g) > 2 ) ) ) ;
   CGAL_precondition( !lBottomFaceExists || (lBottomFaceExists && ( degree(target(qb, g), g) > 2 ) ) ) ;
@@ -858,7 +859,7 @@ collapse_edge(typename boost::graph_traits<Graph>::edge_descriptor v0v1,
 
   if ( lTopFaceExists )
   { 
-    CGAL_precondition( ! internal::is_border(opposite(pt, g),g) ) ; // p-q-t is a face of the mesh
+    CGAL_precondition( ! is_border(opposite(pt, g),g) ) ; // p-q-t is a face of the mesh
     if ( lTopLeftFaceExists )
     {
       //CGAL_ECMS_TRACE(3, "Removing p-t E" << pt.idx() << " (V" 
@@ -885,7 +886,7 @@ collapse_edge(typename boost::graph_traits<Graph>::edge_descriptor v0v1,
 
   if ( lBottomFaceExists )
   {   
-    CGAL_precondition( ! internal::is_border(opposite(qb, g),g) ) ; // p-q-b is a face of the mesh
+    CGAL_precondition( ! is_border(opposite(qb, g),g) ) ; // p-q-b is a face of the mesh
     if ( lBottomRightFaceExists )
     {
       //CGAL_ECMS_TRACE(3, "Removing q-b E" << qb.idx() << " (V" 
@@ -946,8 +947,8 @@ collapse_edge(typename boost::graph_traits<Graph>::edge_descriptor v0v1,
   halfedge_descriptor tq = opposite(next(pq,g),g);
   halfedge_descriptor bp = opposite(next(qp,g),g);
 
-  bool lTopFaceExists         = ! internal::is_border(pq,g) ;
-  bool lBottomFaceExists      = ! internal::is_border(qp,g) ;
+  bool lTopFaceExists         = ! is_border(pq,g) ;
+  bool lBottomFaceExists      = ! is_border(qp,g) ;
 
   vertex_descriptor q = target(pq,g);
   vertex_descriptor p = source(pq,g);
@@ -986,7 +987,7 @@ collapse_edge(typename boost::graph_traits<Graph>::edge_descriptor v0v1,
   if (lTopFaceExists && lBottomFaceExists)
   {
     if ( face(edges_to_erase[0],g) == face(edges_to_erase[1],g)
-         && (! internal::is_border(edges_to_erase[0],g)) )
+         && (! is_border(edges_to_erase[0],g)) )
     {
       // the vertex is of valence 3 and we simply need to remove the vertex
       // and its indicent edges
@@ -1001,11 +1002,11 @@ collapse_edge(typename boost::graph_traits<Graph>::edge_descriptor v0v1,
     }
     else
     {
-      if (!(internal::is_border(edges_to_erase[0],g)))
+      if (!(is_border(edges_to_erase[0],g)))
         join_face(edges_to_erase[0],g);
       else
         remove_face(opposite(edges_to_erase[0],g),g);
-      if (!internal::is_border(edges_to_erase[1],g))
+      if (!is_border(edges_to_erase[1],g))
         join_face(edges_to_erase[1],g);
       else
         remove_face(opposite(edges_to_erase[1],g),g);
@@ -1017,22 +1018,22 @@ collapse_edge(typename boost::graph_traits<Graph>::edge_descriptor v0v1,
   {
       if (lTopFaceExists)
       {
-        if (!(internal::is_border(edges_to_erase[0],g))){
+        if (!(is_border(edges_to_erase[0],g))){
           join_face(edges_to_erase[0],g);
           join_vertex(pq,g);
           return q;
         }
-        bool lQ_Erased = internal::is_border(opposite(next(pq,g),g),g);
+        bool lQ_Erased = is_border(opposite(next(pq,g),g),g);
         remove_face(opposite(edges_to_erase[0],g),g);
         return lQ_Erased?p:q;
       }
 
-      if (! (internal::is_border(edges_to_erase[0],g))){
+      if (! (is_border(edges_to_erase[0],g))){
         join_face(edges_to_erase[0],g);
         join_vertex(qp,g);
         return p;
       }
-      bool lP_Erased= internal::is_border(opposite(next(qp,g),g),g);
+      bool lP_Erased= is_border(opposite(next(qp,g),g),g);
       remove_face(opposite(edges_to_erase[0],g),g);
       return lP_Erased?q:p;
   };
diff --git a/BGL/include/CGAL/boost/graph/Graph_geometry.h b/BGL/include/CGAL/boost/graph/Graph_geometry.h
index edc8cb9d7f9..5aa6851821b 100644
--- a/BGL/include/CGAL/boost/graph/Graph_geometry.h
+++ b/BGL/include/CGAL/boost/graph/Graph_geometry.h
@@ -205,25 +205,7 @@ triangle(const HalfedgeGraph& g,
 }
 
 
-/** 
- * Check if all faces in a \ref FaceListGraph are triangles.
- *
- * @tparam FaceListGraph Must be a model of \ref FaceListGraph.
- *
- * @return true, if all faces are triangles, false otherwise.
- */
-template<typename FaceListGraph>
-bool
-is_pure_triangle(const FaceListGraph& g)
-{
-  typedef boost::graph_traits<FaceListGraph> Traits;
-  typedef typename Traits::face_iterator face_iterator;
-  face_iterator fb, fe;
-  for(boost::tie(fb, fe) = faces(g); fb != fe; ++fb) {
-    if(boost::distance(halfedges_around_face(halfedge(*fb,g), g)) != 3) return false;
-  }
-  return true;
-}
+
 
 /// @}
 
diff --git a/BGL/include/CGAL/boost/graph/graph_traits_PolyMesh_ArrayKernelT.h b/BGL/include/CGAL/boost/graph/graph_traits_PolyMesh_ArrayKernelT.h
index 20fac824a71..377c9617b59 100644
--- a/BGL/include/CGAL/boost/graph/graph_traits_PolyMesh_ArrayKernelT.h
+++ b/BGL/include/CGAL/boost/graph/graph_traits_PolyMesh_ArrayKernelT.h
@@ -26,7 +26,7 @@
 #include <boost/iterator/transform_iterator.hpp>
 
 #include <CGAL/boost/graph/properties_PolyMesh_ArrayKernelT.h>
-#include <CGAL/BGL/OM_iterator_from_circulator.h>
+#include <CGAL/boost/graph/internal/OM_iterator_from_circulator.h>
 #include <CGAL/boost/graph/iterator.h>
 
 #include <OpenMesh/Core/Mesh/PolyMesh_ArrayKernelT.hh>
diff --git a/BGL/include/CGAL/boost/graph/helpers.h b/BGL/include/CGAL/boost/graph/helpers.h
index a484b65b1f8..34abfbb59d7 100644
--- a/BGL/include/CGAL/boost/graph/helpers.h
+++ b/BGL/include/CGAL/boost/graph/helpers.h
@@ -7,15 +7,12 @@
 namespace CGAL {
 
 
-#if 0
-todo: cleanup as this also is in CGAL/BGL/Helpers.h
 
 template <typename FaceGraph>
 bool is_border(typename boost::graph_traits<FaceGraph>::halfedge_descriptor hd, const FaceGraph& g)
 {
   return face(hd,g) == boost::graph_traits<FaceGraph>::null_face();
 }
-#endif
 
  /*!
     returns `true` if there are no 
diff --git a/BGL/include/CGAL/BGL/OM_iterator_from_circulator.h b/BGL/include/CGAL/boost/graph/internal/OM_iterator_from_circulator.h
similarity index 100%
rename from BGL/include/CGAL/BGL/OM_iterator_from_circulator.h
rename to BGL/include/CGAL/boost/graph/internal/OM_iterator_from_circulator.h
diff --git a/BGL/include/CGAL/BGL/Helper.h b/BGL/include/CGAL/boost/graph/internal/helpers.h
similarity index 95%
rename from BGL/include/CGAL/BGL/Helper.h
rename to BGL/include/CGAL/boost/graph/internal/helpers.h
index 0a63ee3e6d1..abb4bedc8e1 100644
--- a/BGL/include/CGAL/BGL/Helper.h
+++ b/BGL/include/CGAL/boost/graph/internal/helpers.h
@@ -1,5 +1,5 @@
-#ifndef CGAL_BGL_HELPERS_H
-#define CGAL_BGL_HELPERS_H
+#ifndef CGAL_BOOST_GRAPH_INTERNAL_HELPERS_H
+#define CGAL_BOOST_GRAPH_INTERNAL_HELPERS_H
 
 #include <boost/graph/graph_traits.hpp>
 #include <boost/optional.hpp>
@@ -7,23 +7,12 @@
 
 namespace CGAL {
 
-// breaks a dependency loop between <CGAL/BGL/Helper.h> and
-// <CGAL/boost/graph/iterator.h>
+// breaks a dependency loop between <CGAL/boost/graph/helpers.h>
+// and <CGAL/boost/graph/iterator.h>
 template <typename Graph> class Halfedge_around_target_iterator;
 
 namespace internal {
 
-
-
-
-template <typename Graph>
-bool
-is_border(typename boost::graph_traits<Graph>::halfedge_descriptor const& h
-          , const Graph& g)
-{
-  return face(h,g) == boost::graph_traits<Graph>::null_face();
-}
-
 template <typename Graph>
 void
 set_border(typename boost::graph_traits<Graph>::halfedge_descriptor const& h
@@ -278,4 +267,4 @@ bool is_valid(const Graph& g)
 } // CGAL
 
 
-#endif /* CGAL_BGL_HELPERS_H */
+#endif // CGAL_BOOST_GRAPH_INTERNAL_HELPERS_H
diff --git a/BGL/include/CGAL/boost/graph/iterator.h b/BGL/include/CGAL/boost/graph/iterator.h
index e8d1195318a..63553ca5314 100644
--- a/BGL/include/CGAL/boost/graph/iterator.h
+++ b/BGL/include/CGAL/boost/graph/iterator.h
@@ -28,7 +28,7 @@
 
 #include <CGAL/assertions.h>
 #include <CGAL/circulator_bases.h>
-#include <CGAL/BGL/Helper.h>
+#include <CGAL/boost/graph/internal/helpers.h>
 
 namespace CGAL {
 
diff --git a/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h b/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h
index 735742cb0bb..319e844d631 100644
--- a/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h
+++ b/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h
@@ -59,13 +59,13 @@ class Polyhedron_edge_index_map_external
   : public boost::put_get_helper<std::size_t, Polyhedron_edge_index_map_external<Polyhedron> >
 {
 public:
-  typedef boost::readable_property_map_tag                              category;
-  typedef std::size_t                                                   value_type;
-  typedef std::size_t                                                   reference;
-  typedef typename boost::graph_traits<Polyhedron>::halfedge_descriptor key_type;
+  typedef boost::readable_property_map_tag                          category;
+  typedef std::size_t                                               value_type;
+  typedef std::size_t                                               reference;
+  typedef typename boost::graph_traits<Polyhedron>::edge_descriptor key_type;
 
   Polyhedron_edge_index_map_external(Polyhedron& p)
-    : map_(std::size_t(-1), num_halfedges(p))
+    : map_(std::size_t(-1), num_halfedges(p)), p(p)
   {
     unsigned int data = 0;
     typename boost::graph_traits<Polyhedron>::edge_iterator it, end;
@@ -75,10 +75,11 @@ public:
     }
   }
 
-  reference operator[](const key_type& k) const { return map_[k]; }
+  reference operator[](const key_type& k) const { return map_[halfedge(k,p)]; }
 private:
   CGAL::Unique_hash_map<typename boost::graph_traits<Polyhedron>::halfedge_descriptor,
                         std::size_t> map_;
+  Polyhedron& p;
 };
 
 template<typename Handle>
diff --git a/BGL/test/BGL/test_Euler_operations.cpp b/BGL/test/BGL/test_Euler_operations.cpp
index 08df2b828f0..9a1833367aa 100644
--- a/BGL/test/BGL/test_Euler_operations.cpp
+++ b/BGL/test/BGL/test_Euler_operations.cpp
@@ -22,8 +22,6 @@ BOOST_AUTO_TEST_CASE_TEMPLATE( join_face_test, T, test_graphs )
   set_halfedge(f.f1, e, f.m);
   CGAL::Euler::join_face(e,f.m);
 
-  CGAL::internal::set_constant_vertex_is_border(f.m);
-
   BOOST_CHECK(CGAL::internal::exact_num_faces(f.m) == 2);
   BOOST_CHECK(CGAL::internal::exact_num_edges(f.m) == 6);
   
diff --git a/BGL/test/BGL/test_Prefix.h b/BGL/test/BGL/test_Prefix.h
index 7819ecf32d8..d277715bf4f 100644
--- a/BGL/test/BGL/test_Prefix.h
+++ b/BGL/test/BGL/test_Prefix.h
@@ -50,7 +50,7 @@ typedef CGAL::Surface_mesh<Point_3> SM;
 typedef OpenMesh::PolyMesh_ArrayKernelT</* MyTraits*/> OMesh;
 #endif
 
-#include <CGAL/BGL/Helper.h>
+#include <CGAL/boost/graph/helpers.h>
 
 // helper to easily define all graph_traits members
 #define CGAL_GRAPH_TRAITS_MEMBERS(T)                                    \
diff --git a/Operations_on_polyhedra/include/CGAL/Polyhedron_slicer_3.h b/Operations_on_polyhedra/include/CGAL/Polyhedron_slicer_3.h
index c479e00dd6d..52078e67d44 100644
--- a/Operations_on_polyhedra/include/CGAL/Polyhedron_slicer_3.h
+++ b/Operations_on_polyhedra/include/CGAL/Polyhedron_slicer_3.h
@@ -24,7 +24,7 @@
 #include <CGAL/AABB_tree.h>
 #include <CGAL/AABB_traits.h>
 #include <CGAL/AABB_halfedge_graph_segment_primitive.h>
-#include <CGAL/boost/graph/halfedge_graph_traits_Polyhedron_3.h>
+#include <CGAL/boost/graph/graph_traits_Polyhedron_3.h>
 
 #include <CGAL/Vector_3.h>
 #include <CGAL/Point_3.h>
@@ -48,7 +48,7 @@ template<class Polyhedron, class Kernel>
 class Polyhedron_slicer_3
 {
 private:
-  typedef AABB_halfedge_graph_segment_primitive<const Polyhedron>    AABB_primitive;
+  typedef AABB_halfedge_graph_segment_primitive<Polyhedron>         AABB_primitive;
   typedef AABB_traits<Kernel, AABB_primitive>                       AABB_traits_;
   typedef AABB_tree<AABB_traits_>                                   AABB_tree_;
 
@@ -59,12 +59,12 @@ private:
   typedef typename Kernel::Segment_3  Segment;
   typedef typename Kernel::Point_3    Point;
 
-  typedef typename Polyhedron::Edge_const_iterator   Edge_const_iterator;
-  typedef typename Polyhedron::Halfedge_const_handle Halfedge_const_handle;
-  typedef typename Polyhedron::Facet_const_handle    Facet_const_handle;
-  typedef typename Polyhedron::Vertex_const_handle   Vertex_const_handle;
-  typedef typename Polyhedron::Halfedge_around_vertex_const_circulator Halfedge_around_vertex_const_circulator;
-  typedef typename Polyhedron::Halfedge_around_facet_const_circulator  Halfedge_around_facet_const_circulator;
+  typedef typename boost::graph_traits<Polyhedron>::edge_iterator   Edge_const_iterator;
+  typedef typename boost::graph_traits<Polyhedron>::halfedge_descriptor Halfedge_const_handle;
+  typedef typename boost::graph_traits<Polyhedron>::face_descriptor    Facet_const_handle;
+  typedef typename boost::graph_traits<Polyhedron>::vertex_descriptor   Vertex_const_handle;
+  typedef Halfedge_around_target_circulator<Polyhedron> Halfedge_around_vertex_const_circulator;
+  typedef Halfedge_around_face_circulator<Polyhedron>  Halfedge_around_facet_const_circulator;
 
   // to unite halfedges under an "edge"
   struct Edge_comparator {
@@ -127,6 +127,7 @@ private:
   typename Kernel::Intersect_3 intersect_3_functor;
   AABB_tree_ tree;
   mutable Node_graph node_graph;
+  Polyhedron& polyhedron;
 
   boost::tuple<Point, Intersection_type, Vertex_const_handle>
   halfedge_intersection(Halfedge_const_handle hf, const Plane& plane) const
@@ -177,11 +178,11 @@ private:
     Vertex_g v_g = boost::add_vertex(node_graph);
     node_graph[v_g].point = v->point();
 
-    Halfedge_around_vertex_const_circulator around_vertex_c = v->vertex_begin();
+    Halfedge_around_vertex_const_circulator around_vertex_c(v,polyhedron), done(around_vertex_c);
     do {
-      edge_node_map[around_vertex_c].put(v_g);
+      edge_node_map[*around_vertex_c].put(v_g);
     } 
-    while(++around_vertex_c != v->vertex_begin());
+    while(++around_vertex_c != done);
   }
   
   void add_intersection_edge_to_facet_neighbors(Halfedge_const_handle hf, Edge_node_map& edge_node_map) const {
@@ -212,11 +213,11 @@ private:
     Node_pair& hf_node_pair = edge_node_map.find(hf)->second;
     Vertex_g v_g = node_graph[hf_node_pair.v1].point == v->point() ? hf_node_pair.v1 : hf_node_pair.v2;// find node containing v
 
-    Halfedge_around_vertex_const_circulator around_vertex_c = v->vertex_begin();
+    Halfedge_around_vertex_const_circulator around_vertex_c(v, polyhedron), done(around_vertex_c);
     do {
-      if(around_vertex_c->is_border()) { continue;} 
-      Node_pair& around_vertex_node_pair = edge_node_map.find(around_vertex_c)->second;
-      Halfedge_around_facet_const_circulator around_facet_c = around_vertex_c->facet_begin();
+      if((*around_vertex_c)->is_border()) { continue;} 
+      Node_pair& around_vertex_node_pair = edge_node_map.find(*around_vertex_c)->second;
+      Halfedge_around_facet_const_circulator around_facet_c(*around_vertex_c,polyhedron), done2(around_facet_c);
       do {
         CGAL_assertion(around_vertex_node_pair.vertex_count != 0);
         if(around_vertex_node_pair.v1 != v_g) {
@@ -226,9 +227,9 @@ private:
           boost::add_edge(around_vertex_node_pair.v2, v_g, node_graph); 
         }
       } 
-      while(++around_facet_c != around_vertex_c->facet_begin());
+      while(++around_facet_c != done2);
     } 
-    while(++around_vertex_c != v->vertex_begin());
+    while(++around_vertex_c != done);
   }
 
   template<class OutputIterator>
@@ -419,9 +420,10 @@ public:
   */
   Polyhedron_slicer_3(const Polyhedron& polyhedron, const Kernel& kernel = Kernel())
   : intersect_3_functor(kernel.intersect_3_object()),
-    tree( undirected_edges(polyhedron).first,
-          undirected_edges(polyhedron).second,
-          polyhedron)
+    tree( edges(polyhedron).first,
+          edges(polyhedron).second,
+          polyhedron),
+    polyhedron(const_cast<Polyhedron&>(polyhedron))
   { }
 
   /**