diff --git a/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h b/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
index f6435051313..76f7468d469 100644
--- a/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
+++ b/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
@@ -143,10 +143,9 @@ public:
   }
 
   // endpoints of constraints are unremovable
-  // vertices which have more than 1 constraint passing through are unremovable
+  // vertices which are not endpoint and have != 2 incident constrained edges are unremovable
   void initialize_unremovable()
   {
-    std::unordered_map<Vertex_handle, int> degrees;
     Constraint_iterator cit = pct.constraints_begin(), e = pct.constraints_end();
     for(; cit!=e; ++cit){
       Constraint_id cid = *cit;
@@ -155,18 +154,30 @@ public:
       (*it)->set_removable(false);
       ++it;
       for(; it != ite; ++it){
-        Vertex_handle vh = *it;
-        ++degrees[vh];
-        if((boost::next(it) != ite) && (boost::prior(it)== boost::next(it))){
-          (*it)->set_removable(false);
+        if(boost::next(it) != ite){
+          Vertex_handle vp = *boost::prior(it), vn = *boost::next(it);
+          if(vp == vn){
+            (*it)->set_removable(false);
+          }
         }
       }
       it = boost::prior(it);
       (*it)->set_removable(false);
     }
 
+    std::unordered_map<Vertex_handle, int> degrees;
+    for (Constrained_edges_iterator it = pct.constrained_edges_begin(); it != pct.constrained_edges_end(); ++it) {
+      Edge e = *it;
+      Face_handle fh = e.first;
+      int ei = e.second;
+      Vertex_handle vh = fh->vertex(pct.cw(ei));
+      ++degrees[vh];
+      vh = fh->vertex(pct.ccw(ei));
+      ++degrees[vh];
+    }
+
     for(Finite_vertices_iterator it = pct.finite_vertices_begin(); it != pct.finite_vertices_end(); ++it){
-      if( it->is_removable() && (degrees[it] > 1) ){
+      if( it->is_removable() && (degrees[it] != 2) ){
         it->set_removable(false);
       }
     }