diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Detail/Edge_collapse.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Detail/Edge_collapse.h
index ac1d04667f9..11511803df7 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Detail/Edge_collapse.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Detail/Edge_collapse.h
@@ -60,7 +60,7 @@ public:
   typedef GetPlacement_     GetPlacement ;
   typedef ShouldStop_       ShouldStop ;
   typedef EdgeIdxMap_       EdgeIdxMap ;
-  typedef EdgeCachedPtrMap_ EdgeCachedPtrMap ;
+  typedef EdgeCachedPtrMap_ EdgeExtraPtrMap ;
   typedef VertexIsFixedMap_ VertexIsFixedMap ;
   typedef VisitorT_         VisitorT ;
   
@@ -170,7 +170,7 @@ public:
               , GetPlacement     const& aGetPlacement
               , ShouldStop       const& aShouldStop 
               , EdgeIdxMap       const& aEdge_idx_map 
-              , EdgeCachedPtrMap const& aEdge_cached_ptr_map 
+              , EdgeExtraPtrMap const& aEdge_extra_ptr_map 
               , VertexIsFixedMap const& aVertex_is_fixed_map 
               , VisitorT*               aVisitor
               ) ;
@@ -212,12 +212,12 @@ private:
     return is_border(aEdge) || is_border(opposite_edge(aEdge,mSurface)) ;
   }    
   
-  Edge_data_ptr get_data ( edge_descriptor const& aEdge ) const { return static_cast<Edge_data_ptr>(get(Edge_cached_ptr_map,aEdge)) ; }
+  Edge_data_ptr get_data ( edge_descriptor const& aEdge ) const { return static_cast<Edge_data_ptr>(get(Edge_extra_ptr_map,aEdge)) ; }
   
   void set_data ( edge_descriptor const& aEdge, Edge_data_ptr aData )
   {
-    put(Edge_cached_ptr_map,aEdge,aData) ;
-    put(Edge_cached_ptr_map,opposite_edge(aEdge,mSurface),aData) ;
+    put(Edge_extra_ptr_map,aEdge,aData) ;
+    put(Edge_extra_ptr_map,opposite_edge(aEdge,mSurface),aData) ;
   }
   
   tuple<const_vertex_descriptor,const_vertex_descriptor> get_vertices( const_edge_descriptor const& aEdge ) const
@@ -326,7 +326,7 @@ private:
   GetPlacement     const& Get_placement ;
   ShouldStop       const& Should_stop ;
   EdgeIdxMap       const& Edge_idx_map ;
-  EdgeCachedPtrMap const& Edge_cached_ptr_map ;
+  EdgeExtraPtrMap  const& Edge_extra_ptr_map ;
   VertexIsFixedMap const& Vertex_is_fixed_map ;
   VisitorT*               Visitor ;
   
diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Detail/Edge_collapse_impl.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Detail/Edge_collapse_impl.h
index 299f7e16bac..980395b53c4 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Detail/Edge_collapse_impl.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Detail/Edge_collapse_impl.h
@@ -23,15 +23,15 @@ CGAL_BEGIN_NAMESPACE
 namespace Triangulated_surface_mesh { namespace Simplification { namespace Edge_collapse 
 {
 
-template<class M,class P,class D,class C,class V,class S, class I, class P, class F, class R>
-EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::EdgeCollapse( TSM&                    aSurface
+template<class M,class P,class D,class C,class V,class S, class I, class X, class F, class R>
+EdgeCollapse<M,P,D,C,V,S,I,X,F,R>::EdgeCollapse( TSM&                    aSurface
                                                , Params           const* aParams
                                                , SetCollapseData  const& aSet_collapse_data
                                                , GetCost          const& aGet_cost
                                                , GetPlacement     const& aGet_placement
                                                , ShouldStop       const& aShould_stop
                                                , EdgeIdxMap       const& aEdge_idx_map 
-                                               , EdgeCachedPtrMap const& aEdge_cached_ptr_map 
+                                               , EdgeExtraPtrMap  const& aEdge_extra_ptr_map 
                                                , VertexIsFixedMap const& aVertex_is_fixed_map 
                                                , VisitorT*               aVisitor 
                                                )
@@ -44,14 +44,14 @@ EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::EdgeCollapse( TSM&                    aSurfac
   ,Get_placement      (aGet_placement)
   ,Should_stop        (aShould_stop) 
   ,Edge_idx_map       (Edge_idx_map)
-  ,Edge_cached_ptr_map(Edge_cached_ptr_map)
+  ,Edge_extra_ptr_map (Edge_extra_ptr_map)
   ,Vertex_is_fixed_map(Vertex_is_fixed_map)
   ,Visitor            (aVisitor)
   
 {
   CGAL_expensive_precondition( is_valid_triangulated_surface_mesh(mSurface) );
   
-  CGAL_expensive_precondition( check_max_id(mSurface) );
+  CGAL_expensive_precondition( check_max_id() );
 
   mPQ.reset( new PQ (num_edges(aSurface), Compare_cost(this), Edge_idx_map) ) ;
   
@@ -70,8 +70,8 @@ EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::EdgeCollapse( TSM&                    aSurfac
 #endif
 }
 
-template<class M,class P,class D,class C,class V,class S, class I, class P, class F, class R>
-int EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::run()
+template<class M,class P,class D,class C,class V,class S, class I, class X, class F, class R>
+int EdgeCollapse<M,P,D,C,V,S,I,X,F,R>::run()
 {
   if ( Visitor )
     Visitor->OnStarted(mSurface);
@@ -94,8 +94,8 @@ int EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::run()
   return r ;
 }
 
-template<class M,class P,class D,class C,class V,class S, class I, class P, class F, class R>
-void EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::Collect()
+template<class M,class P,class D,class C,class V,class S, class I, class X, class F, class R>
+void EdgeCollapse<M,P,D,C,V,S,I,X,F,R>::Collect()
 {
   CGAL_TSMS_TRACE(0,"Collecting edges...");
 
@@ -151,8 +151,8 @@ void EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::Collect()
   CGAL_TSMS_TRACE(0,"Initial edge count: " << mInitialEdgeCount ) ;
 }
 
-template<class M,class P,class D,class C,class V,class S, class I, class P, class F, class R>
-void EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::Loop()
+template<class M,class P,class D,class C,class V,class S, class I, class X, class F, class R>
+void EdgeCollapse<M,P,D,C,V,S,I,X,F,R>::Loop()
 {
   CGAL_TSMS_TRACE(0,"Collapsing edges...") ;
 
@@ -212,8 +212,8 @@ void EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::Loop()
 //
 // The link conidition is as follows: for every vertex 'k' adjacent to both 'p and 'q', "p,k,q" is a facet of the mesh.
 //
-template<class M,class P,class D,class C,class V,class S, class I, class P, class F, class R>
-bool EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::Is_collapsable( edge_descriptor const& aEdgePQ )
+template<class M,class P,class D,class C,class V,class S, class I, class X, class F, class R>
+bool EdgeCollapse<M,P,D,C,V,S,I,X,F,R>::Is_collapsable( edge_descriptor const& aEdgePQ )
 {
   bool rR = true ;
 
@@ -304,8 +304,8 @@ bool EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::Is_collapsable( edge_descriptor const& a
   return rR ;
 }
 
-template<class M,class P,class D,class C,class V,class S, class I, class P, class F, class R>
-typename EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::vertex_descriptor EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::Collapse( edge_descriptor const& aEdgePQ )
+template<class M,class P,class D,class C,class V,class S, class I, class X, class F, class R>
+typename EdgeCollapse<M,P,D,C,V,S,I,X,F,R>::vertex_descriptor EdgeCollapse<M,P,D,C,V,S,I,X,F,R>::Collapse( edge_descriptor const& aEdgePQ )
 {
   CGAL_TSMS_TRACE(1,"S" << mStep << ". Collapsig " << edge_to_string(aEdgePQ) ) ;
   
@@ -415,8 +415,8 @@ typename EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::vertex_descriptor EdgeCollapse<M,P,D
   return rResult ;
 }
 
-template<class M,class P,class D,class C,class V,class S, class I, class P, class F, class R>
-void EdgeCollapse<M,P,D,C,V,S,I,P,F,R>::Update_neighbors( vertex_descriptor const& aKeptV )
+template<class M,class P,class D,class C,class V,class S, class I, class X, class F, class R>
+void EdgeCollapse<M,P,D,C,V,S,I,X,F,R>::Update_neighbors( vertex_descriptor const& aKeptV )
 {
   CGAL_TSMS_TRACE(3,"Updating cost of neighboring edges..." ) ;
 
diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Edge_collapse.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Edge_collapse.h
index 87c89dfa215..589f736613c 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Edge_collapse.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Edge_collapse.h
@@ -39,7 +39,7 @@ namespace Triangulated_surface_mesh { namespace Simplification { namespace Edge_
 //
 //   For each non-fixed edge in the mesh, a "collapse data" record is constructed by calling the user-supplied function
 //   "SetCollapseData".
-//   The edge is then associated with its collapse data via the "EdgeCachedPtrMap" property map.
+//   The edge is then associated with its collapse data via the "EdgeExtraPtrMap" property map.
 //
 //   The user-supplied function "GetCost" is called, for each non-fixed edge.
 //   This function returns a value which defines the collapsing cost of the edge. Edges with a lower cost are collapsed first.
@@ -66,7 +66,7 @@ template<class TSM
         ,class GetNewVertexPoint
         ,class ShouldStop
         ,class EdgeIdxMap
-        ,class EdgeCachedPtrMap
+        ,class EdgeExtraPtrMap
         ,class VertexIsFixedMap
         ,class Visitor
         >
@@ -77,7 +77,7 @@ int edge_collapse ( TSM&                     aSurface
                   , GetNewVertexPoint const& aGet_new_vertex_point
                   , ShouldStop        const& aShould_stop
                   , EdgeIdxMap        const& aEdge_idx_map 
-                  , EdgeCachedPtrMap  const& aEdge_cached_ptr_map
+                  , EdgeExtraPtrMap   const& aEdge_extra_ptr_map
                   , VertexIsFixedMap  const& aVertex_is_fixed_map
                   , Visitor*                 aVisitor = 0
                   ) 
@@ -89,7 +89,7 @@ int edge_collapse ( TSM&                     aSurface
                       ,GetNewVertexPoint
                       ,ShouldStop
                       ,EdgeIdxMap
-                      ,EdgeCachedPtrMap
+                      ,EdgeExtraPtrMap
                       ,VertexIsFixedMap
                       ,Visitor
                       >
@@ -103,7 +103,7 @@ int edge_collapse ( TSM&                     aSurface
                      ,aGet_new_vertex_point
                      ,aShould_stop
                      ,aEdge_idx_map 
-                     ,aEdge_cached_ptr_map
+                     ,aEdge_extra_ptr_map
                      ,aVertex_is_fixed_map
                      ,aVisitor
                      ) ;
diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Edge_stored_index_map.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Edge_stored_index_map.h
index eefdd725ec3..d5fbae2782b 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Edge_stored_index_map.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Edge_stored_index_map.h
@@ -23,7 +23,7 @@
 namespace boost
 {
 
-template<class Graph>
+template<class Graph_>
 class Edge_stored_index_map : public put_get_helper< typename Graph_::size_type, Edge_stored_index_map<Graph_> >
 {
 private:
@@ -32,7 +32,7 @@ private:
 
 public:
 
-  typedef typename G::size_type size_type ;
+  typedef typename Graph::size_type size_type ;
   
   typedef readable_property_map_tag                           category;
   typedef size_type                                           value_type;
diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Vertex_stored_is_fixed_map.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Vertex_stored_is_fixed_map.h
index 14c71d2cda1..27828100bbe 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Vertex_stored_is_fixed_map.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Vertex_stored_is_fixed_map.h
@@ -37,7 +37,7 @@ public:
   typedef bool                                                  reference;
   typedef typename graph_traits<Graph const>::vertex_descriptor key_type;
 
-  reference operator[](key_type const& v) const { v->is_fixed() ; }
+  reference operator[](key_type const& v) const { return v->is_fixed() ; }
 };
 
         
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/edge_collapse_test.cpp b/Surface_mesh_simplification/test/Surface_mesh_simplification/edge_collapse_test.cpp
index feeec2ac1d0..be16e07858a 100644
--- a/Surface_mesh_simplification/test/Surface_mesh_simplification/edge_collapse_test.cpp
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/edge_collapse_test.cpp
@@ -73,15 +73,15 @@ struct My_vertex : public HalfedgeDS_vertex_base<Refs,Tag_true,Point>
 {
   typedef HalfedgeDS_vertex_base<Refs,Tag_true,Point> Base ;
  
-  My_vertex() : ID(-1), IsFixed(false) {} 
-  My_vertex( Point p ) : Base(p), ID(-1), IsFixed(false) {}
+  My_vertex() : mID(-1), mIsFixed(false) {} 
+  My_vertex( Point p ) : Base(p), mID(-1), mIsFixed(false) {}
 
-  int id() const { return ID ; }
+  int id() const { return mID ; }
   
-  bool is_fixed() const { return IsFixed ; }
+  bool is_fixed() const { return mIsFixed ; }
     
-  int  ID; 
-  bool IsFixed ;
+  int  mID; 
+  bool mIsFixed ;
 } ;
 
 template <class Refs, class Traits>
@@ -89,17 +89,16 @@ struct My_halfedge : public HalfedgeDS_halfedge_base<Refs>
 { 
   My_halfedge() 
    : 
-    ID(-1) 
-   , extra_pointer(0)
+     mID(-1) 
+   , mExtraPointer(0)
   {}
  
-  int id() const { return ID ; }
+  int id() const { return mID ; }
 
-  void* extra_pointer() { return extra_pointer ; }
+  void*& extra_pointer() { return mExtraPointer ; }
     
-  int ID; 
-  
-  void* extra_pointer ;
+  int   mID; 
+  void* mExtraPointer ;
 };
 
 template <class Refs, class Traits>
@@ -107,12 +106,12 @@ struct My_face : public HalfedgeDS_face_base<Refs,Tag_true,typename Traits::Plan
 {
   typedef HalfedgeDS_face_base<Refs,Tag_true,typename Traits::Plane_3> Base ;
   
-  My_face() : ID(-1) {}
-  My_face( typename Traits::Plane_3 plane ) : Base(plane) {}
+  My_face() : mID(-1) {}
+  My_face( typename Traits::Plane_3 plane ) : Base(plane), mID(-1) {}
   
-  int id() const { return ID ; }
+  int id() const { return mID ; }
   
-  int ID; 
+  int mID; 
 };
 
 struct My_items : public Polyhedron_items_3 
@@ -520,15 +519,15 @@ bool Test ( int aStopA, int aStopR, bool aJustPrintSurfaceData, string aName, Me
         
           int lVertexID = 0 ;
           for ( Polyhedron::Vertex_iterator vi = lP.vertices_begin(); vi != lP.vertices_end() ; ++ vi )
-            vi->ID = lVertexID ++ ;    
+            vi->mID = lVertexID ++ ;    
           
           int lHalfedgeID = 0 ;
           for ( Polyhedron::Halfedge_iterator hi = lP.halfedges_begin(); hi != lP.halfedges_end() ; ++ hi )
-            hi->ID = lHalfedgeID++ ;    
+            hi->mID = lHalfedgeID++ ;    
           
           int lFacetID = 0 ;
           for ( Polyhedron::Facet_iterator fi = lP.facets_begin(); fi != lP.facets_end() ; ++ fi )
-            fi->ID = lFacetID ++ ;    
+            fi->mID = lFacetID ++ ;    
       
 #ifdef AUDIT
           sAuditData .clear();
@@ -558,7 +557,9 @@ bool Test ( int aStopA, int aStopR, bool aJustPrintSurfaceData, string aName, Me
           Dummy_params lDummy_params;
           LT_params    lLT_params ; 
 
-          Edge_index_map<Polyhedron> P_Edge_index_map ;
+          Edge_stored_index_map        <Polyhedron> edge_index_map ;
+          Edge_stored_extra_pointer_map<Polyhedron> edge_extra_pointer_map ;
+          Vertex_stored_is_fixed_map   <Polyhedron> vertex_is_fixed_map ;
           
           Visitor lVisitor(lRequestedEdgeCount) ;
       
@@ -579,6 +580,9 @@ bool Test ( int aStopA, int aStopR, bool aJustPrintSurfaceData, string aName, Me
                                    ,get_MP_cost
                                    ,get_MP_placement
                                    ,should_stop
+                                   ,edge_index_map 
+                                   ,edge_extra_pointer_map 
+                                   ,vertex_is_fixed_map 
                                    ,&lVisitor
                                    );
                   break ;                 
@@ -591,6 +595,9 @@ bool Test ( int aStopA, int aStopR, bool aJustPrintSurfaceData, string aName, Me
                                    ,get_cached_cost
                                    ,get_MP_placement
                                    ,should_stop
+                                   ,edge_index_map 
+                                   ,edge_extra_pointer_map 
+                                   ,vertex_is_fixed_map 
                                    ,&lVisitor
                                    );
                   break ;                  
@@ -603,6 +610,9 @@ bool Test ( int aStopA, int aStopR, bool aJustPrintSurfaceData, string aName, Me
                                    ,get_cached_cost
                                    ,get_cached_placement
                                    ,should_stop
+                                   ,edge_index_map 
+                                   ,edge_extra_pointer_map 
+                                   ,vertex_is_fixed_map 
                                    ,&lVisitor
                                    );
                   break ;                  
@@ -623,6 +633,9 @@ bool Test ( int aStopA, int aStopR, bool aJustPrintSurfaceData, string aName, Me
                                    ,get_LT_cost
                                    ,get_LT_placement
                                    ,should_stop
+                                   ,edge_index_map 
+                                   ,edge_extra_pointer_map 
+                                   ,vertex_is_fixed_map 
                                    ,&lVisitor
                                    );
                   break ;                 
@@ -635,6 +648,9 @@ bool Test ( int aStopA, int aStopR, bool aJustPrintSurfaceData, string aName, Me
                                    ,get_cached_cost
                                    ,get_LT_placement
                                    ,should_stop
+                                   ,edge_index_map 
+                                   ,edge_extra_pointer_map 
+                                   ,vertex_is_fixed_map 
                                    ,&lVisitor
                                    );
                   break ;                  
@@ -647,6 +663,9 @@ bool Test ( int aStopA, int aStopR, bool aJustPrintSurfaceData, string aName, Me
                                    ,get_cached_cost
                                    ,get_cached_placement
                                    ,should_stop
+                                   ,edge_index_map 
+                                   ,edge_extra_pointer_map 
+                                   ,vertex_is_fixed_map 
                                    ,&lVisitor
                                    );
                   break ;