diff --git a/BGL/include/CGAL/boost/graph/graph_traits_Polyhedron_3.h b/BGL/include/CGAL/boost/graph/graph_traits_Polyhedron_3.h
index 66895fa773c..ef8fe02fcd3 100644
--- a/BGL/include/CGAL/boost/graph/graph_traits_Polyhedron_3.h
+++ b/BGL/include/CGAL/boost/graph/graph_traits_Polyhedron_3.h
@@ -17,8 +17,8 @@
 //
 // Author(s): Andreas Fabri <andreas.fabri@geometryfactory.com>, Fernando Cacciola <fernando.cacciola@gmail.com>
 
-#ifndef CGAL_BOOST_GRAPH_POLYHEDRON_GRAPH_TRAITS_H
-#define CGAL_BOOST_GRAPH_POLYHEDRON_GRAPH_TRAITS_H
+#ifndef CGAL_BOOST_GRAPH_GRAPH_TRAITS_POLYHEDRON_3_H
+#define CGAL_BOOST_GRAPH_GRAPH_TRAITS_POLYHEDRON_3_H
 
 #include <CGAL/boost/graph/graph_traits_HalfedgeDS.h>
 
@@ -208,4 +208,4 @@ out_edges( typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_descr
 
 #undef CGAL_HDS_
 
-#endif // CGAL_BOOST_GRAPH_POLYHEDRON_GRAPH_TRAITS_H
+#endif // CGAL_BOOST_GRAPH_GRAPH_TRAITS_POLYHEDRON_3_H
diff --git a/BGL/include/CGAL/boost/graph/halfedge_graph_traits.h b/BGL/include/CGAL/boost/graph/halfedge_graph_traits.h
index 6741a99198c..53bbd6e5783 100644
--- a/BGL/include/CGAL/boost/graph/halfedge_graph_traits.h
+++ b/BGL/include/CGAL/boost/graph/halfedge_graph_traits.h
@@ -24,7 +24,7 @@
 
 CGAL_BEGIN_NAMESPACE
 
-template<class T> struct Halfedge_graph_traits ;
+template<class T> struct halfedge_graph_traits ;
 
 CGAL_END_NAMESPACE
 
diff --git a/BGL/include/CGAL/boost/graph/halfedge_graph_traits_HalfedgeDS.h b/BGL/include/CGAL/boost/graph/halfedge_graph_traits_HalfedgeDS.h
index 76d75bc2dbd..073b97e28ff 100644
--- a/BGL/include/CGAL/boost/graph/halfedge_graph_traits_HalfedgeDS.h
+++ b/BGL/include/CGAL/boost/graph/halfedge_graph_traits_HalfedgeDS.h
@@ -17,8 +17,8 @@
 //
 // Author(s): Fernando Cacciola <fernando.cacciola@gmail.com>, Andreas Fabri <andreas.fabri@geometryfactory.com>
 
-#ifndef CGAL_BOOST_GRAPH_GRAPH_TRAITS_HALFEDGEDS_H
-#define CGAL_BOOST_GRAPH_GRAPH_TRAITS_HALFEDGEDS_H
+#ifndef CGAL_BOOST_GRAPH_HALFEDGE_GRAPH_TRAITS_HALFEDGEDS_H
+#define CGAL_BOOST_GRAPH_HALFEDGE_GRAPH_TRAITS_HALFEDGEDS_H
 
 #include <boost/config.hpp>
 #include <boost/iterator_adaptors.hpp>
@@ -85,4 +85,4 @@ public :
 
 CGAL_END_NAMESPACE
 
-#endif // CGAL_BOOST_GRAPH_GRAPH_TRAITS_HALFEDGEDS_H
+#endif // CGAL_BOOST_GRAPH_HALFEDGE_GRAPH_TRAITS_HALFEDGEDS_H
diff --git a/BGL/include/CGAL/boost/graph/halfedge_graph_traits_Polyhedron_3.h b/BGL/include/CGAL/boost/graph/halfedge_graph_traits_Polyhedron_3.h
index 9947e046738..3b99097320d 100644
--- a/BGL/include/CGAL/boost/graph/halfedge_graph_traits_Polyhedron_3.h
+++ b/BGL/include/CGAL/boost/graph/halfedge_graph_traits_Polyhedron_3.h
@@ -17,11 +17,11 @@
 //
 // Author(s) : Fernando Caccciola <fernando.cacciola@gmail.com>
 
-#ifndef CGAL_BOOST_GRAPH_POLYHEDRON_HALFEDGE_GRAPH_TRAITS_H
-#define CGAL_BOOST_GRAPH_POLYHEDRON_HALFEDGE_GRAPH_TRAITS_H
+#ifndef CGAL_BOOST_GRAPH_HALFEDGE_GRAPH_TRAITS_POLYHEDRON_3_H
+#define CGAL_BOOST_GRAPH_HALFEDGE_GRAPH_TRAITS_POLYHEDRON_3_H
 
 #include <CGAL/HalfedgeDS_items_decorator.h>
-#include <CGAL/boost/graph/graph_traits_Polyhedron.h>
+#include <CGAL/boost/graph/graph_traits_Polyhedron_3.h>
 #include <CGAL/boost/graph/halfedge_graph_traits.h>
 #include <CGAL/boost/graph/halfedge_graph_traits_HalfedgeDS.h>
 
@@ -37,18 +37,18 @@ CGAL_BEGIN_NAMESPACE
 // Const versions
 // 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-struct Halfedge_graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const > 
+struct halfedge_graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const > 
   : CGAL::HDS_halfedge_graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>
 {
 };
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-inline std::pair<typename Halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> const>::undirected_edge_iterator
-                ,typename Halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> const>::undirected_edge_iterator 
+inline std::pair<typename halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> const>::undirected_edge_iterator
+                ,typename halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> const>::undirected_edge_iterator 
                 >  
 undirected_edges( Polyhedron_3<Gt,I,HDS,A> const& p )
 {
-  typedef typename Halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> const>::undirected_edge_iterator Iter;
+  typedef typename halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> const>::undirected_edge_iterator Iter;
   return std::make_pair( Iter(p.edges_begin()), Iter(p.edges_end()) );
 }
 
@@ -105,18 +105,18 @@ next_edge_cw( typename boost::graph_traits< Polyhedron_3<Gt,I,HDS,A> const>::edg
 // 
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-struct Halfedge_graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> > 
+struct halfedge_graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> > 
   : CGAL::HDS_halfedge_graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >
 {
 };
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-inline std::pair<typename Halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> >::undirected_edge_iterator
-                ,typename Halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> >::undirected_edge_iterator 
+inline std::pair<typename halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> >::undirected_edge_iterator
+                ,typename halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> >::undirected_edge_iterator 
                 >  
 undirected_edges( Polyhedron_3<Gt,I,HDS,A>& p )
 {
-  typedef typename Halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> >::undirected_edge_iterator Iter;
+  typedef typename halfedge_graph_traits< Polyhedron_3<Gt,I,HDS,A> >::undirected_edge_iterator Iter;
   return std::make_pair( Iter(p.edges_begin()), Iter(p.edges_end()) );
 }
 
@@ -176,4 +176,4 @@ CGAL_END_NAMESPACE
 
 #undef CGAL_HDS_
 
-#endif // CGAL_BOOST_GRAPH_POLYHEDRON_HALFEDGE_GRAPH_TRAITS_H
+#endif // CGAL_BOOST_GRAPH_HALFEDGE_GRAPH_TRAITS_POLYHEDRON_3_H
diff --git a/BGL/include/CGAL/boost/graph/named_function_params.h b/BGL/include/CGAL/boost/graph/named_function_params.h
index 8a0317a8c78..ffdc001ea41 100644
--- a/BGL/include/CGAL/boost/graph/named_function_params.h
+++ b/BGL/include/CGAL/boost/graph/named_function_params.h
@@ -50,6 +50,7 @@
 
 CGAL_BEGIN_NAMESPACE
 
+  enum vertex_is_fixed_t             { vertex_is_fixed             } ;  
   enum set_cache_policy_t            { set_cache_policy            } ;
   enum get_cost_policy_t             { get_cost_policy             } ;
   enum get_cost_policy_params_t      { get_cost_policy_params      } ;
@@ -63,8 +64,12 @@ CGAL_BEGIN_NAMESPACE
     typedef Base next_type;
     typedef Tag tag_type;
     typedef T value_type;
+
+    cgal_bgl_named_params() { }
+
     cgal_bgl_named_params(T v) : m_value(v) { }
     cgal_bgl_named_params(T v, const Base& b) : Base(b), m_value(v) { }
+
     T m_value;
 
     template <typename IndexMap>
@@ -213,34 +218,34 @@ CGAL_BEGIN_NAMESPACE
   }
   
   template <typename GetCost>
-  cgal_bgl_named_params<GetCost, get_cost_policy_t, self>
-  get_cost(const GetCost& p) const 
+  cgal_bgl_named_params<GetCost, get_cost_policy_t>
+  get_cost(const GetCost& p)
   {
-    typedef cgal_bgl_named_params<GetCost, get_cost_policy_t, self> Params;
+    typedef cgal_bgl_named_params<GetCost, get_cost_policy_t> Params;
      return Params(p);
   }
 
   template <typename GetCostParams>
-  cgal_bgl_named_params<GetCostParams, get_cost_policy_params_t, self>
-  get_cost_params(const GetCostParams& p) const 
+  cgal_bgl_named_params<GetCostParams, get_cost_policy_params_t>
+  get_cost_params(const GetCostParams& p)
   {
-    typedef cgal_bgl_named_params<GetCostParams, get_cost_policy_params_t, self> Params;
+    typedef cgal_bgl_named_params<GetCostParams, get_cost_policy_params_t> Params;
     return Params(p);
   }
   
   template <typename GetPlacement>
-  cgal_bgl_named_params<GetPlacement, get_placement_policy_t, self>
-  get_placement(const GetPlacement& p) const 
+  cgal_bgl_named_params<GetPlacement, get_placement_policy_t>
+  get_placement(const GetPlacement& p) 
   {
-    typedef cgal_bgl_named_params<GetPlacement, get_placement_policy_t, self> Params;
+    typedef cgal_bgl_named_params<GetPlacement, get_placement_policy_t> Params;
     return Params(p);
   }
   
   template <typename GetPlacementParams>
-  cgal_bgl_named_params<GetPlacementParams, get_placement_policy_params_t, self>
-  get_placement_params(const GetPlacementParams& p) const 
+  cgal_bgl_named_params<GetPlacementParams, get_placement_policy_params_t>
+  get_placement_params(const GetPlacementParams& p)
   {
-    typedef cgal_bgl_named_params<GetPlacementParams, get_placement_policy_params_t, self> Params;
+    typedef cgal_bgl_named_params<GetPlacementParams, get_placement_policy_params_t> Params;
     return Params(p);
   }
   
@@ -255,7 +260,7 @@ CGAL_BEGIN_NAMESPACE
       boost::property_value< cgal_bgl_named_params<T1,Tag1,Base>, Tag2>::type T2;
     T2* t2 = 0;
     typedef boost::detail::property_value_dispatch<match> Dispatcher;
-    return boost::Dispatcher::const_get_value(p, t2, tag2);
+    return Dispatcher::const_get_value(p, t2, tag2);
   }
 
 CGAL_END_NAMESPACE
diff --git a/BGL/include/CGAL/boost/graph/properties.h b/BGL/include/CGAL/boost/graph/properties.h
index ac56ae9a46a..96797496a14 100644
--- a/BGL/include/CGAL/boost/graph/properties.h
+++ b/BGL/include/CGAL/boost/graph/properties.h
@@ -27,9 +27,11 @@
 
 CGAL_BEGIN_NAMESPACE
 
-enum edge_is_border_t   { edge_is_border   } ;
-enum vertex_is_border_t { vertex_is_border } ;
-enum vertex_point_t     { vertex_point     } ;
+enum vertex_is_border_t      { vertex_is_border } ;
+enum vertex_point_t          { vertex_point     } ;
+enum vertex_external_index_t { vertex_external_index } ;
+enum edge_is_border_t        { edge_is_border   } ;
+enum edge_external_index_t   { edge_external_index } ;
 
 CGAL_END_NAMESPACE
 
diff --git a/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h b/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h
index ecb70be21c7..7d8a82a12c0 100644
--- a/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h
+++ b/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h
@@ -17,11 +17,11 @@
 //
 // Author(s): Andreas Fabri <andreas.fabri@geometryfactory.com>, Fernando Cacciola <fernando.cacciola@gmail.com>
 
-#ifndef CGAL_BOOST_GRAPH_PROPERTIES_POLYHEDRON_H
-#define CGAL_BOOST_GRAPH_PROPERTIES_POLYHEDRON_H
+#ifndef CGAL_BOOST_GRAPH_PROPERTIES_POLYHEDRON_3_H
+#define CGAL_BOOST_GRAPH_PROPERTIES_POLYHEDRON_3_H
 
 #include <CGAL/boost/graph/properties.h>
-#include <CGAL/boost/graph/graph_traits_Polyhedron.h>
+#include <CGAL/boost/graph/graph_traits_Polyhedron_3.h>
 #include <CGAL/Unique_hash_map.h>
 
 #ifndef CGAL_CFG_NO_TMPL_IN_TMPL_PARAM
@@ -136,7 +136,7 @@ public:
 
   Polyhedron_vertex_point_map( Polyhedron& ) {}
 
-  reference operator[](key_type const& v) const { return e->point(); }
+  reference operator[](key_type const& v) const { return v->point(); }
 };
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
@@ -159,11 +159,11 @@ public:
 
   Polyhedron_vertex_point_const_map( Polyhedron const& ) {}
 
-  reference operator[](key_type const& v) const { return e->point(); }
+  reference operator[](key_type const& v) const { return v->point(); }
 };
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-class Polyhedron_vertex_index_map_stored : public boost::put_get_helper<strd::size_t, Polyhedron_vertex_index_map_stored<Gt, I, HDS, A> >
+class Polyhedron_vertex_index_map_stored : public boost::put_get_helper<std::size_t, Polyhedron_vertex_index_map_stored<Gt, I, HDS, A> >
 {
 private:
 
@@ -195,7 +195,7 @@ public:
   typedef std::size_t                                                       reference;
   typedef typename boost::graph_traits<Polyhedron const>::vertex_descriptor key_type;
 
-  Polyhedron_edge_index_map_external( Polyhedron const& p) 
+  Polyhedron_vertex_index_map_external( Polyhedron const& p) 
     : map( p.vertices_begin(),p.vertices_end(),0,std::size_t(-1),p.size_of_vertices() )
   {}
 
@@ -233,6 +233,17 @@ struct Polyhedron_property_map<boost::edge_index_t>
   };
 };
 
+template <>
+struct Polyhedron_property_map<edge_external_index_t> 
+{
+  template<class Gt, class I, CGAL_HDS_PARAM_, class A>
+  struct bind_ 
+  {
+    typedef Polyhedron_edge_index_map_external<Gt,I,HDS,A> type;
+    typedef Polyhedron_edge_index_map_external<Gt,I,HDS,A> const_type;
+  };
+};
+
 template <>
 struct Polyhedron_property_map<edge_is_border_t> 
 {
@@ -244,17 +255,6 @@ struct Polyhedron_property_map<edge_is_border_t>
   };
 };
 
-template <>
-struct Polyhedron_property_map<vertex_is_border_t> 
-{
-  template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-  struct bind_ 
-  {
-    typedef Polyhedron_vertex_is_border_map<Gt,I,HDS,A> type;
-    typedef Polyhedron_vertex_is_border_map<Gt,I,HDS,A> const_type;
-  };
-};
-
 template <>
 struct Polyhedron_property_map<vertex_point_t> 
 {
@@ -277,6 +277,17 @@ struct Polyhedron_property_map<boost::vertex_index_t>
   };
 };
 
+template <>
+struct Polyhedron_property_map<vertex_external_index_t> 
+{
+  template<class Gt, class I, CGAL_HDS_PARAM_, class A>
+  struct bind_ 
+  {
+    typedef Polyhedron_vertex_index_map_external<Gt,I,HDS,A> type;
+    typedef Polyhedron_vertex_index_map_external<Gt,I,HDS,A> const_type;
+  };
+};
+
 CGAL_END_NAMESPACE
 
 namespace boost
@@ -308,9 +319,9 @@ CGAL::Polyhedron_edge_index_map_stored<Gt,I,HDS,A> get( edge_index_t, CGAL::Poly
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
 inline 
-CGAL::Polyhedron_vertex_is_border_map<Gt,I,HDS,A> get(CGAL::vertex_is_border_t, CGAL::Polyhedron_3<Gt,I,HDS,A> const& p) 
+CGAL::Polyhedron_edge_index_map_external<Gt,I,HDS,A> get( CGAL::edge_external_index_t, CGAL::Polyhedron_3<Gt,I,HDS,A> const& p) 
 {
-  CGAL::Polyhedron_vertex_is_border_map<Gt,I,HDS,A> m(p);
+  CGAL::Polyhedron_edge_index_map_external<Gt,I,HDS,A> m(p);
   return m;
 }
 
@@ -338,6 +349,14 @@ CGAL::Polyhedron_vertex_index_map_stored<Gt,I,HDS,A> get(vertex_index_t, CGAL::P
   return m;
 }
 
+template<class Gt, class I, CGAL_HDS_PARAM_, class A>
+inline 
+CGAL::Polyhedron_vertex_index_map_external<Gt,I,HDS,A> get(CGAL::vertex_external_index_t, CGAL::Polyhedron_3<Gt,I,HDS,A> const& p) 
+{
+  CGAL::Polyhedron_vertex_index_map_external<Gt,I,HDS,A> m(p);
+  return m;
+}
+
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A, class Tag>
 struct property_map<CGAL::Polyhedron_3<Gt,I,HDS,A>, Tag> 
@@ -390,4 +409,4 @@ struct vertex_property_type<CGAL::Polyhedron_3<Gt,I,HDS,A> >
 
 #undef CGAL_HDS_PARAM_
 
-#endif // CGAL_BOOST_GRAPH_POLYHEDRON_BGL_PROPERTIES_H
+#endif // CGAL_BOOST_GRAPH_PROPERTIES_POLYHEDRON_3_H