diff --git a/Surface_mesh_simplification/doc_tex/Surface_mesh_simplification_ref/edge_collapse.tex b/Surface_mesh_simplification/doc_tex/Surface_mesh_simplification_ref/edge_collapse.tex
index b84972ad4b7..463b141c6f8 100644
--- a/Surface_mesh_simplification/doc_tex/Surface_mesh_simplification_ref/edge_collapse.tex
+++ b/Surface_mesh_simplification/doc_tex/Surface_mesh_simplification_ref/edge_collapse.tex
@@ -80,17 +80,28 @@ and whose \ccc{value_type} is
 \subsubsection{\ccc{edge_index_map(EdgeIndexMap edge_idx_map)} }
 
 Maps each {\em directed} edge in the surface into an integer number
-in the range \ccc{[0,num_edge(surface)}.\\
-\ccc{EdgeIndexMap} must be a 
+in the range \ccc{[0,num_edge(surface)}.
+
+\ccc{EdgeIndexMap} must be a either a
 \ccAnchor{http://www.boost.org/libs/property_map/ReadablePropertyMap.html}{ReadablePropertyMap} 
 whose \ccc{key_type} is
 \ccc{boost::graph_traits<EdgeCollapsableMesh const>::edge_descriptor}
 and whose \ccc{value_type} is 
-\ccc{boost::graph_traits<EdgeCollapsableMesh>::size_type}
+\ccc{boost::graph_traits<EdgeCollapsableMesh>::size_type},
+or the same as \ccc{EdgeCollapsableMesh}, in which case 
+\ccc{edge_idx_map} is in fact the surface to be simplified.
+
+If \ccc{edge_idx_map} is a valid property map, it is
+passed as the argument, but if it is the surface, a
+temporary property map is passed instead.
+Such temporary map externaly relates each edge 
+in the surface (\ccc{edge_idx_map}) to its index,
+without requiring any user intervention or any special
+provision in the surface edge.
 
 \textbf{Default}: the property map obtained by calling \ccc{get(edge_index,surface)}, which 
 in turn requires the surface edge class to have a member function 
-\ccc{edges_size_type id() const;}.
+\ccc{int id() const;}.
 
 \subsubsection{\ccc{edge_is_border_map(EdgeIsBorderMap edge_border_map)} }
 
diff --git a/Surface_mesh_simplification/examples/Surface_mesh_simplification/Edge_collapse_fully_cached_polyhedron.cpp b/Surface_mesh_simplification/examples/Surface_mesh_simplification/Edge_collapse_fully_cached_polyhedron.cpp
index 224ddca1296..34578496a5f 100644
--- a/Surface_mesh_simplification/examples/Surface_mesh_simplification/Edge_collapse_fully_cached_polyhedron.cpp
+++ b/Surface_mesh_simplification/examples/Surface_mesh_simplification/Edge_collapse_fully_cached_polyhedron.cpp
@@ -52,7 +52,7 @@ int main( int argc, char** argv )
     SMS::edge_collapse(surface
                       ,SMS::Count_stop_condition<Surface>(1000)             
                        
-                      ,SMS::external_edge_index_map(surface)
+                      ,SMS::edge_index_map(surface)
                       
                       .SMS::set_cache    (SMS::LindstromTurk_set_cost_and_placement_cache<Surface>())
                       .SMS::get_cost     (SMS::Cached_cost     <Surface>())
@@ -91,7 +91,7 @@ int main( int argc, char** argv )
     SMS::edge_collapse(surface
                       ,SMS::Count_stop_condition<Surface>(1000)         
                        
-                      ,SMS::external_edge_index_map(surface)
+                      ,SMS::edge_index_map(surface)
                       
                       .SMS::set_cache    (set_full_cache)
                       .SMS::get_cost     (SMS::Cached_cost     <Surface>())
diff --git a/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_enriched_polyhedron.cpp b/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_enriched_polyhedron.cpp
index 3153277b2e7..3abc9bd732e 100644
--- a/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_enriched_polyhedron.cpp
+++ b/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_enriched_polyhedron.cpp
@@ -33,7 +33,7 @@ struct My_items : public CGAL::Polyhedron_items_3
     };
     template < class Refs, class Traits>
     struct Halfedge_wrapper { 
-        typedef CGAL::HalfedgeDS_halfedge_base_with_id<Refs> Halfedge;
+        typedef CGAL::HalfedgeDS_halfedge_max_base_with_id<Refs> Halfedge;
     };
     template < class Refs, class Traits>
     struct Face_wrapper {
diff --git a/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_polyhedron.cpp b/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_polyhedron.cpp
index 810414f9444..4604c9ff54e 100644
--- a/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_polyhedron.cpp
+++ b/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_polyhedron.cpp
@@ -38,7 +38,7 @@ int main( int argc, char** argv )
   // while third argument could have been omited, as shown in the next example.
   int r = SMS::edge_collapse(surface
                             ,stop_policy                 
-                            ,SMS::external_edge_index_map(surface)
+                            ,SMS::edge_index_map(surface)
                             );
   
   // === CONCRETE USAGE EXAMPLE ENDS HERE ===
diff --git a/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_visited_polyhedron.cpp b/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_visited_polyhedron.cpp
index 7b9598d3c05..fcb19a2cbe2 100644
--- a/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_visited_polyhedron.cpp
+++ b/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_visited_polyhedron.cpp
@@ -123,7 +123,7 @@ int main( int argc, char** argv )
 
   int r = SMS::edge_collapse(surface
                              ,SMS::Count_ratio_stop_condition<Surface>(0.10) 
-                             ,SMS::external_edge_index_map(surface)
+                             ,SMS::edge_index_map(surface)
                              .SMS::visitor(&vis)
                              );
 
diff --git a/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_with_fixed_vertices_enriched_polyhedron.cpp b/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_with_fixed_vertices_enriched_polyhedron.cpp
index bed563cea1e..cdf52ce18b4 100644
--- a/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_with_fixed_vertices_enriched_polyhedron.cpp
+++ b/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_with_fixed_vertices_enriched_polyhedron.cpp
@@ -53,7 +53,7 @@ struct My_items : public CGAL::Polyhedron_items_3
     };
     template < class Refs, class Traits>
     struct Halfedge_wrapper { 
-        typedef CGAL::HalfedgeDS_halfedge_base_with_id<Refs> Halfedge;
+        typedef CGAL::HalfedgeDS_halfedge_max_base_with_id<Refs> Halfedge;
     };
     template < class Refs, class Traits>
     struct Face_wrapper {
@@ -90,7 +90,7 @@ int main( int argc, char** argv )
                              ,SMS::Count_ratio_stop_condition<Surface>(0.10) 
                              ,SMS.vertex_is_fixed_map(CGAL::Vertex_is_fixed_map_stored<Surface>())
                              // The edge_index_map parameter is ommited becasue
-                             // the halfedge in this polyhedron support the stored id().
+                             // the halfedge in this polyhedron supports the stored id().
                              );
            
   // === CONCRETE USAGE EXAMPLE ENDS HERE ===
diff --git a/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_with_fixed_vertices_polyhedron.cpp b/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_with_fixed_vertices_polyhedron.cpp
index 89e46a94570..8cf9f361509 100644
--- a/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_with_fixed_vertices_polyhedron.cpp
+++ b/Surface_mesh_simplification/examples/Surface_mesh_simplification/LT_edge_collapse_with_fixed_vertices_polyhedron.cpp
@@ -47,7 +47,7 @@ int main( int argc, char** argv )
  
   int r = SMS::edge_collapse(surface
                             ,SMS::Count_ratio_stop_condition<Surface>(0.10)
-                            ,SMS::external_edge_index_map(surface)
+                            ,SMS::edge_index_map(surface)
                             .SMS::vertex_is_fixed_map(boost::make_assoc_property_map(is_vertex_fixed))
                             );
   
diff --git a/Surface_mesh_simplification/examples/Surface_mesh_simplification/MP_edge_collapse_polyhedron.cpp b/Surface_mesh_simplification/examples/Surface_mesh_simplification/MP_edge_collapse_polyhedron.cpp
index 99c5a36dd77..87a2490d216 100644
--- a/Surface_mesh_simplification/examples/Surface_mesh_simplification/MP_edge_collapse_polyhedron.cpp
+++ b/Surface_mesh_simplification/examples/Surface_mesh_simplification/MP_edge_collapse_polyhedron.cpp
@@ -40,7 +40,7 @@ int main( int argc, char** argv )
   int r = SMS::edge_collapse(surface
                             ,SMS::Count_ratio_stop_condition<Surface>(0.10)
                              
-                            ,SMS::external_edge_index_map(surface)
+                            ,SMS::edge_index_map(surface)
                             
                             .SMS::set_cache    (SMS::Set_no_cache      <Surface>())
                             .SMS::get_cost     (SMS::Edge_length_cost  <Surface>())