diff --git a/BGL/include/CGAL/boost/graph/graph_traits_HalfedgeDS.h b/BGL/include/CGAL/boost/graph/graph_traits_HalfedgeDS.h
index cb3e8beed39..f989dd19e3d 100644
--- a/BGL/include/CGAL/boost/graph/graph_traits_HalfedgeDS.h
+++ b/BGL/include/CGAL/boost/graph/graph_traits_HalfedgeDS.h
@@ -233,7 +233,6 @@ struct HDS_graph_traits
 public :
   
   struct HDS_graph_traversal_category : public virtual boost::bidirectional_graph_tag,
-					//                                        public virtual boost::adjacency_graph_tag,
                                         public virtual boost::vertex_list_graph_tag,
                                         public virtual boost::edge_list_graph_tag
   {};
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 b6c90ce3bbe..048078f5dd1 100644
--- a/BGL/include/CGAL/boost/graph/graph_traits_Polyhedron_3.h
+++ b/BGL/include/CGAL/boost/graph/graph_traits_Polyhedron_3.h
@@ -31,45 +31,21 @@
 #endif
 
 
+//
+// NOTE: The BGL algorithms are NOT const-correct: i.e., they take a "G const&" 
+// but instantiate "graph_traits<G>" instead of "graph_traits<G const>"
+// This is known Boost bug which will eventually be fixed, but in the meantime we need
+// to coerce both const and non-const specializations.
+// That is, HDS_graph_traits<G const> is really the same as HDS_graph_traits<G>
+// so graph_traits<G> is also the same as graph_traits<G const>.
+// Therefore, while, for instance, "graph_traits<G const>::vertex_descriptor"
+// is conceptually const-correct, it actually corresponds to the non-const handle,
+// hence the const_cast<> used below in the functions implementation.
+//
+
 namespace boost
 { 
 
- 
-template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertices_size_type
-num_vertices(const CGAL::Polyhedron_3<Gt,I,HDS,A>& p)
-{
-  return p.size_of_vertices();
-}
-
-template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::edges_size_type 
-num_edges(const CGAL::Polyhedron_3<Gt,I,HDS,A>& p)
-{
-  return p.size_of_halfedges() ;
-}
-
-template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::degree_size_type
-degree(typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_descriptor v, const CGAL::Polyhedron_3<Gt,I,HDS,A>&)
-{
-  return v->vertex_degree() * 2 ;
-}
-
-template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::degree_size_type
-out_degree(typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_descriptor v, const CGAL::Polyhedron_3<Gt,I,HDS,A>&)
-{
-  return v->vertex_degree();
-}
-
-template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::degree_size_type
-in_degree(typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_descriptor v,  const CGAL::Polyhedron_3<Gt,I,HDS,A>&)
-{
-  return v->vertex_degree();
-}
-
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
 struct graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >
    : CGAL::HDS_graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >
@@ -77,67 +53,109 @@ struct graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >
 
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_descriptor
-source(typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::edge_descriptor e, const CGAL::Polyhedron_3<Gt,I,HDS,A> & )
+struct graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const >
+   : CGAL::HDS_graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> > // See NOTE above!
+{};
+
+ 
+template<class Gt, class I, CGAL_HDS_PARAM_, class A>
+typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertices_size_type
+num_vertices(const CGAL::Polyhedron_3<Gt,I,HDS,A>& p)
+{
+  return p.size_of_vertices();
+}
+
+template<class Gt, class I, CGAL_HDS_PARAM_, class A>
+typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::edges_size_type 
+num_edges(const CGAL::Polyhedron_3<Gt,I,HDS,A>& p)
+{
+  return p.size_of_halfedges() ;
+}
+
+template<class Gt, class I, CGAL_HDS_PARAM_, class A>
+typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::degree_size_type
+degree(typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertex_descriptor v, const CGAL::Polyhedron_3<Gt,I,HDS,A>&)
+{
+  return v->vertex_degree() * 2 ;
+}
+
+template<class Gt, class I, CGAL_HDS_PARAM_, class A>
+typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::degree_size_type
+out_degree(typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertex_descriptor v, const CGAL::Polyhedron_3<Gt,I,HDS,A>&)
+{
+  return v->vertex_degree();
+}
+
+template<class Gt, class I, CGAL_HDS_PARAM_, class A>
+typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::degree_size_type
+in_degree(typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertex_descriptor v,  const CGAL::Polyhedron_3<Gt,I,HDS,A>&)
+{
+  return v->vertex_degree();
+}
+
+
+template<class Gt, class I, CGAL_HDS_PARAM_, class A>
+typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertex_descriptor
+source(typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::edge_descriptor e, const CGAL::Polyhedron_3<Gt,I,HDS,A> & )
 {
   return e->opposite()->vertex();
 }
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_descriptor
-target(typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::edge_descriptor e, const CGAL::Polyhedron_3<Gt,I,HDS,A> & )
+typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertex_descriptor
+target(typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::edge_descriptor e, const CGAL::Polyhedron_3<Gt,I,HDS,A> & )
 {
   return e->vertex();
 }
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-inline std::pair<typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_iterator
-                ,typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_iterator 
+inline std::pair<typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertex_iterator
+                ,typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertex_iterator 
                 >  
 vertices( const CGAL::Polyhedron_3<Gt,I,HDS,A>& p)
 {
-  typedef typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_iterator Iter;
+  typedef typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertex_iterator Iter;
   CGAL::Polyhedron_3<Gt,I,HDS,A>& ncp = const_cast<CGAL::Polyhedron_3<Gt,I,HDS,A>&>(p);
   return std::make_pair( Iter(ncp.vertices_begin()), Iter(ncp.vertices_end()) );
 }
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-inline std::pair<typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::edge_iterator
-                ,typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::edge_iterator 
+inline std::pair<typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::edge_iterator
+                ,typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::edge_iterator 
                 >  
 edges( const CGAL::Polyhedron_3<Gt,I,HDS,A>& p)
 {
-  typedef typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::edge_iterator Iter;
+  typedef typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::edge_iterator Iter;
   CGAL::Polyhedron_3<Gt,I,HDS,A>& ncp = const_cast<CGAL::Polyhedron_3<Gt,I,HDS,A>&>(p);
   return std::make_pair( Iter(ncp.halfedges_begin()), Iter(ncp.halfedges_end()) );
 }
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-inline std::pair<typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::in_edge_iterator
-                ,typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::in_edge_iterator
+inline std::pair<typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::in_edge_iterator
+                ,typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::in_edge_iterator
                 >  
-in_edges( typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_descriptor u, const CGAL::Polyhedron_3<Gt,I,HDS,A>& g)
+in_edges( typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertex_descriptor u, const CGAL::Polyhedron_3<Gt,I,HDS,A>& g)
 {
   typename CGAL::Polyhedron_3<Gt,I,HDS,A>::Halfedge_around_vertex_circulator ec = u->vertex_begin();
-  typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::edges_size_type in_deg = in_degree(u,g);
-  typedef typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::in_edge_iterator Iter;
+  typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::edges_size_type in_deg = in_degree(u,g);
+  typedef typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::in_edge_iterator Iter;
   return std::make_pair( Iter(ec), Iter(ec,in_deg) );
 }
 
 template<class Gt, class I, CGAL_HDS_PARAM_, class A>
-inline std::pair<typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::out_edge_iterator
-                ,typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::out_edge_iterator 
+inline std::pair<typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::out_edge_iterator
+                ,typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::out_edge_iterator 
                 >  
-out_edges( typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::vertex_descriptor u, const CGAL::Polyhedron_3<Gt,I,HDS,A>& g)
+out_edges( typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::vertex_descriptor u, const CGAL::Polyhedron_3<Gt,I,HDS,A>& g)
 {
   typename CGAL::Polyhedron_3<Gt,I,HDS,A>::Halfedge_around_vertex_circulator ec = u->vertex_begin();
-  typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::edges_size_type out_deg = out_degree(u,g);
-  typedef typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> >::out_edge_iterator Iter;
+  typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::edges_size_type out_deg = out_degree(u,g);
+  typedef typename graph_traits< CGAL::Polyhedron_3<Gt,I,HDS,A> const>::out_edge_iterator Iter;
   return std::make_pair( Iter(ec), Iter(ec,out_deg) );
 }
         
 } // namespace boost
 
-#undef CGAL_HDS_
+#undef CGAL_HDS_PARAM_
 
 #endif // CGAL_BOOST_GRAPH_GRAPH_TRAITS_POLYHEDRON_3_H
diff --git a/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h b/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h
index 64e0dcf8395..d892ee1cf05 100644
--- a/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h
+++ b/BGL/include/CGAL/boost/graph/properties_Polyhedron_3.h
@@ -44,7 +44,7 @@ public:
   typedef boost::readable_property_map_tag                                category;
   typedef double                                                          value_type;
   typedef double                                                          reference;
-  typedef typename boost::graph_traits<Polyhedron>::edge_descriptor key_type;
+  typedef typename boost::graph_traits<Polyhedron const>::edge_descriptor key_type;
 
   Polyhedron_edge_weight_map( Polyhedron const& ) {}
 
@@ -66,7 +66,7 @@ public:
   typedef boost::readable_property_map_tag                                category;
   typedef bool                                                            value_type;
   typedef bool                                                            reference;
-  typedef typename boost::graph_traits<Polyhedron>::edge_descriptor key_type;
+  typedef typename boost::graph_traits<Polyhedron const>::edge_descriptor key_type;
 
   Polyhedron_edge_is_border_map( Polyhedron const& ) {}
 
@@ -85,7 +85,7 @@ 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>::edge_descriptor key_type;
+  typedef typename boost::graph_traits<Polyhedron const>::edge_descriptor key_type;
 
   Polyhedron_edge_index_map_stored( Polyhedron const& ) {}
 
@@ -104,16 +104,23 @@ 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>::edge_descriptor key_type;
+  typedef typename boost::graph_traits<Polyhedron const>::edge_descriptor key_type;
 
   Polyhedron_edge_index_map_external( Polyhedron const& p) 
-    : map( p.halfedges_begin(),p.halfedges_end(),0,std::size_t(-1),p.size_of_halfedges() )
+    : map( remove_const(p).halfedges_begin()
+         , remove_const(p).halfedges_end()
+         , 0
+         , std::size_t(-1)
+         , p.size_of_halfedges() 
+         )
   {}
 
   reference operator[](key_type const& e) const { return map[e]; }
   
 private:
 
+  static Polyhedron& remove_const ( Polyhedron const& p ) { return const_cast<Polyhedron&>(p) ; }
+  
   CGAL::Unique_hash_map<key_type,std::size_t> map ;
 };
 
@@ -155,7 +162,7 @@ public:
   typedef boost::readable_property_map_tag                                  category;
   typedef Point_3                                                           value_type;
   typedef Point_3 const&                                                    reference;
-  typedef typename boost::graph_traits<Polyhedron>::vertex_descriptor key_type;
+  typedef typename boost::graph_traits<Polyhedron const>::vertex_descriptor key_type;
 
   Polyhedron_vertex_point_const_map( Polyhedron const& ) {}
 
@@ -174,7 +181,7 @@ 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>::vertex_descriptor key_type;
+  typedef typename boost::graph_traits<Polyhedron const>::vertex_descriptor key_type;
 
   Polyhedron_vertex_index_map_stored( Polyhedron const& ) {}
 
@@ -193,16 +200,23 @@ 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>::vertex_descriptor key_type;
+  typedef typename boost::graph_traits<Polyhedron const>::vertex_descriptor key_type;
 
   Polyhedron_vertex_index_map_external( Polyhedron const& p) 
-    : map( p.vertices_begin(),p.vertices_end(),0,std::size_t(-1),p.size_of_vertices() )
+    : map( remove_const(p).vertices_begin()
+         , remove_const(p).vertices_end()
+         , 0
+         , std::size_t(-1)
+         , p.size_of_vertices() 
+         )
   {}
 
   reference operator[](key_type const& v) const { return map[v]; }
   
 private:
 
+  static Polyhedron& remove_const ( Polyhedron const& p ) { return const_cast<Polyhedron&>(p) ; }
+  
   CGAL::Unique_hash_map<key_type,std::size_t> map ;
 };