diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_functionalities.h b/Combinatorial_map/include/CGAL/Combinatorial_map_functionalities.h
index 3503720430d..87160240682 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_functionalities.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_functionalities.h
@@ -152,6 +152,10 @@ namespace CGAL {
       m_map.free_mark(marktemp);
       m_map.display_darts(std::cout);
 #endif
+
+      m_map.display_darts(std::cout);
+
+      assert(are_paths_valid());
     }
     
     ~Combinatorial_map_tools()
@@ -168,10 +172,12 @@ namespace CGAL {
       {
         if (!m_original_map.is_marked(path[i], m_mark_T))
         {
-          res.push_back(get_first_dart_of_the_path(path[i]));
-          res.push_back(get_second_dart_of_the_path(path[i]));
+          res.push_back(get_first_dart_of_the_path(path[i]), false);
+          res.push_back(get_second_dart_of_the_path(path[i]), false);
         }
       }
+      res.update_is_closed();
+      assert(res.is_closed());
       assert(res.is_valid());
       return res;
     }
@@ -402,20 +408,20 @@ namespace CGAL {
           //         <<m_map.darts().index(p.second)<<": "<<std::flush;
 
           //std::cout<<m_map.darts().index(p.first)<<"; "<<std::flush;
-          p.first=m_map.template beta<0>(p.first);
+          // p.first=m_map.template beta<0>(p.first);
           Dart_const_handle initdart=p.first;
           while (m_map.is_marked(p.first, toremove))
           {
-            p.first=m_map.template beta<2, 0>(p.first);
+            p.first=m_map.template beta<2, 1>(p.first);
             //std::cout<<m_map.darts().index(p.first)<<"; "<<std::flush;
             assert(p.first!=initdart);
           }
           //std::cout<<std::endl;
-          p.second=m_map.template beta<0>(p.second);
+          // p.second=m_map.template beta<0>(p.second);
           initdart=p.second;
           while (m_map.is_marked(p.second, toremove))
           {
-            p.second=m_map.template beta<2, 0>(p.second);
+            p.second=m_map.template beta<2, 1>(p.second);
             //std::cout<<m_map.darts().index(p.second)<<"; "<<std::flush;
             assert(p.second!=initdart);
           }
@@ -465,8 +471,8 @@ namespace CGAL {
         std::pair<Dart_const_handle, Dart_const_handle>& p=itp->second;
         //std::cout<<"Pair: "<<m_map.darts().index(p.first)<<", "
         //         <<m_map.darts().index(p.second)<<std::flush;
-        p.first=m_map.template beta<1>(p.first);
-        p.second=m_map.template beta<1,2>(p.second);
+        p.first=m_map.template beta<0, 2>(p.first);
+        p.second=m_map.template beta<0>(p.second);
         //std::cout<<" -> "<<m_map.darts().index(p.first)<<", "
         //         <<m_map.darts().index(p.second)<<std::endl;
         // WRONG ASSERTS assert(p.first!=p.second);
diff --git a/Combinatorial_map/include/CGAL/Path_generators.h b/Combinatorial_map/include/CGAL/Path_generators.h
index ef99f457567..84fa1ea660c 100644
--- a/Combinatorial_map/include/CGAL/Path_generators.h
+++ b/Combinatorial_map/include/CGAL/Path_generators.h
@@ -475,6 +475,9 @@ void generate_random_closed_path(Path& p, std::size_t nb,
   { CGAL::unmark_cell<typename Path::Map, 2>(p.get_map(), *it, amark); }
 
   p.get_map().free_mark(amark);
+
+  p.update_is_closed(); // TODO we can avoid that because we know that we generated a closed path (to do so, we need a method that put p.is_closed to true)
+  assert(p.is_closed());
 }
 
 } // namespace CGAL
diff --git a/Combinatorial_map/include/CGAL/Path_on_surface.h b/Combinatorial_map/include/CGAL/Path_on_surface.h
index a7b13761225..4981d4d3379 100644
--- a/Combinatorial_map/include/CGAL/Path_on_surface.h
+++ b/Combinatorial_map/include/CGAL/Path_on_surface.h
@@ -439,6 +439,10 @@ public:
   {
     if (is_empty()) return false;
 
+#ifndef NDEBUG
+    bool is_even=length()%2;
+#endif // NDEBUG
+
     Self new_path(m_map);
     bool positive=false;
     std::size_t begin, end;
@@ -471,6 +475,9 @@ public:
           { copy_rest_of_path(end+1, length(), new_path); }
 
           swap(new_path);
+
+          assert(length()%2==is_even); // bracket flattening is supposed to preserve length parity
+
           return true;
         }
       }
@@ -495,25 +502,43 @@ public:
     bool res=false;
     std::size_t i;
     std::size_t lastturn=m_path.size()-(is_closed()?0:1);
-    Self new_path(m_map);
-    for (i=0; i<lastturn; )
+    for (i=0; !res && i<lastturn; ++i)
     {
       if (m_path[i]==m_map.template beta<2>(m_path[next_index(i)]))
-      {
-        i+=2;
-        res=true;
-      }
-      else
-      {
-        new_path.push_back(m_path[i]); // We copy this dart
-        ++i;
-      }
+      { res=true; }
+    }
+
+    if (!res)
+    { return false; }
+
+#ifndef NDEBUG
+    bool is_even=length()%2;
+#endif // NDEBUG
+
+    --i; // Because we did a ++ before to leave the loop
+    // Here there is a spur at position i in the path
+    Self new_path(m_map);
+
+    // Special case, the spur is between last dart of the path and the first dart
+    if (is_closed() && i==m_path.size()-1)
+    {
+      copy_rest_of_path(1, m_path.size()-1, new_path); // copy path between 1 and m_path.size()-2
+    }
+    else
+    { // Otherwise copy darts before the spur
+      if (i>0)
+      { copy_rest_of_path(0, i, new_path); } // copy path between 0 and i-1
+
+      // and the darts after
+      if (i+2<m_path.size())
+      { copy_rest_of_path(i+2, m_path.size(), new_path); } // copy path between 0 and m_path.size()-1
     }
-    if (i==m_path.size()-1)
-    { new_path.push_back(m_path[m_path.size()-1]); } // we copy the last dart
 
     swap(new_path);
-    return res;
+
+    assert(length()%2==is_even); // spur rremoval is supposed to preserve length parity
+
+    return true;
   }
 
   // Simplify the path by removing all spurs
@@ -615,6 +640,7 @@ public:
       /* else
       { assert(false); } // We think (?) that this case is not possible */
     }
+
   }
 
   void push_l_shape_cycle_2()
@@ -651,6 +677,10 @@ public:
     if (length()==2)
     { return push_l_shape_2darts(); }
 
+#ifndef NDEBUG
+    bool is_even=length()%2;
+#endif // NDEBUG
+
     std::size_t begin, middle, end;
     std::size_t lastturn=m_path.size()-(is_closed()?0:1);
     std::size_t next_turn;
@@ -728,6 +758,9 @@ public:
           { copy_rest_of_path(end+1, length(), new_path); }
 
           swap(new_path);
+
+          assert(length()%2==is_even); // push lshape is supposed to preserve length parity (maybe preserve length ?? TODO check)
+
           return true;
 
         }
@@ -743,7 +776,7 @@ public:
     while(right_push_one_step())
     { res=true;
       
-      std::cout<<"RP "; display();  display_pos_and_neg_turns();
+      std::cout<<"PUSH "; display();  display_pos_and_neg_turns();
       std::cout<<std::endl;
     }
     return res;
diff --git a/Linear_cell_complex/examples/Linear_cell_complex/compute_example.cpp b/Linear_cell_complex/examples/Linear_cell_complex/compute_example.cpp
index 1ab3d6d580b..75e3b66f7f1 100644
--- a/Linear_cell_complex/examples/Linear_cell_complex/compute_example.cpp
+++ b/Linear_cell_complex/examples/Linear_cell_complex/compute_example.cpp
@@ -18,7 +18,7 @@ typedef CGAL::Linear_cell_complex_for_combinatorial_map<2,3> LCC_3_cmap;
 typedef CGAL::Linear_cell_complex_for_generalized_map<2,3> LCC_3_gmap;
 
 #define NB_TESTS 25 // 0 ... 24
-int nbtests=0;
+static int nbtests=0;
 
 enum Transformation // enum for the type of transformations
 {
@@ -27,7 +27,7 @@ enum Transformation // enum for the type of transformations
   FULL_SIMPLIFICATION
 };
 
-unsigned int starting_seed;
+static unsigned int starting_seed;
 
 ///////////////////////////////////////////////////////////////////////////////
 void transform_path(CGAL::Path_on_surface<LCC_3_cmap>& path, Transformation t,
@@ -164,7 +164,7 @@ bool unit_test_canonize(std::vector<CGAL::Path_on_surface<LCC_3_cmap> >& paths,
     std::cout<<"."<<std::flush;
 #endif
 
-    for (int i=0; i<paths.size(); ++i)
+    for (unsigned int i=0; i<paths.size(); ++i)
     {
       if (draw)
       {
@@ -524,7 +524,7 @@ bool test_double_torus_quad(bool draw, int testtorun)
   std::vector<CGAL::Path_on_surface<LCC_3_cmap> > transformed_paths;
 
   // Test 23 (3 g1 cycles)
-  for (int i=0; i<3; ++i)
+  for (unsigned int i=0; i<3; ++i)
   {
     paths.push_back(CGAL::Path_on_surface<LCC_3_cmap>(lcc));
     CGAL::generate_g1_double_torus(paths[i], i);
@@ -559,11 +559,11 @@ bool test_double_torus_quad(bool draw, int testtorun)
     generate_random_closed_path(p, random.get_int(5, 20), random); // random path, length between 30 and 500
     // p.close();
     paths.push_back(p);
-    /* update_path_randomly(p, random);
-    paths.push_back(p);
     update_path_randomly(p, random);
+    paths.push_back(p);
+    /* update_path_randomly(p, random);
     paths.push_back(p);  */
-    for (int i=0; i<paths.size(); ++i)
+    for (unsigned int i=0; i<paths.size(); ++i)
     {
       transformed_paths.push_back
           (cmt2.transform_original_path_into_quad_surface(paths[i]));
@@ -672,7 +672,7 @@ int main(int argc, char** argv)
   CGAL::Random r; // Used when user do not provide its own seed.
   starting_seed=r.get_int(0,INT_MAX);
 
-  for (int i=1; i<argc; ++i)
+  for (unsigned int i=1; i<(unsigned int)argc; ++i)
   {
     arg=argv[i];
     if (arg=="-draw")