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 198928e2145..9e7fe636d01 100644
--- a/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
+++ b/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
@@ -154,8 +154,11 @@ public:
       (*it)->set_removable(false);
       ++it;
       for(; it != ite; ++it){
-        if((std::next(it) != ite) && (std::prev(it)== std::next(it))){
-          (*it)->set_removable(false);
+        if(std::next(it) != ite){
+          Vertex_handle vp = *std::prev(it), vn = *std::next(it);
+          if(vp == vn){
+            (*it)->set_removable(false);
+          }
         }
       }
       it = std::prev(it);
diff --git a/Polyline_simplification_2/test/Polyline_simplification_2/CMakeLists.txt b/Polyline_simplification_2/test/Polyline_simplification_2/CMakeLists.txt
index 6099c152bd8..a2d190b3529 100644
--- a/Polyline_simplification_2/test/Polyline_simplification_2/CMakeLists.txt
+++ b/Polyline_simplification_2/test/Polyline_simplification_2/CMakeLists.txt
@@ -8,5 +8,6 @@ project(Polyline_simplification_2_Tests)
 find_package(CGAL REQUIRED)
 
 create_single_source_cgal_program( "issue-5774.cpp" )
+create_single_source_cgal_program( "issue-8735.cpp" )
 create_single_source_cgal_program( "simplify_polygon_test.cpp" )
 create_single_source_cgal_program( "simplify_polyline_with_duplicate_points.cpp" )
diff --git a/Polyline_simplification_2/test/Polyline_simplification_2/issue-8735.cpp b/Polyline_simplification_2/test/Polyline_simplification_2/issue-8735.cpp
new file mode 100644
index 00000000000..11f2cb75643
--- /dev/null
+++ b/Polyline_simplification_2/test/Polyline_simplification_2/issue-8735.cpp
@@ -0,0 +1,34 @@
+#include <CGAL/Constrained_Delaunay_triangulation_2.h>
+#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
+#include <CGAL/Polyline_simplification_2/simplify.h>
+#include <vector>
+
+namespace PS = CGAL::Polyline_simplification_2;
+typedef CGAL::Exact_predicates_exact_constructions_kernel K;
+typedef PS::Vertex_base_2<K> Vb;
+typedef CGAL::Constrained_triangulation_face_base_2<K> Fb;
+typedef CGAL::Triangulation_data_structure_2<Vb, Fb> TDS;
+typedef CGAL::Constrained_Delaunay_triangulation_2<K, TDS , CGAL::Exact_predicates_tag> CDT;
+typedef CGAL::Constrained_triangulation_plus_2<CDT>     CT;
+typedef CT::Point                            Point;
+typedef PS::Stop_above_cost_threshold        Stop;
+typedef PS::Squared_distance_cost            Cost;
+
+int main()
+{
+  double tolerance = 100;
+  CT ct;
+  std::vector<CT::Point> pts;
+
+  pts.push_back(CT::Point(0, 0));
+  pts.push_back(CT::Point(2, 0));
+  pts.push_back(CT::Point(1, 0));
+  pts.push_back(CT::Point(3, 0));
+  pts.push_back(CT::Point(4, 1));
+  tolerance = 100;
+  ct.insert_constraint(pts.begin(), pts.end(), false);
+
+  PS::simplify(ct, Cost(), Stop(tolerance * tolerance), false);
+
+  return 0;
+}
diff --git a/STL_Extension/include/CGAL/Skiplist.h b/STL_Extension/include/CGAL/Skiplist.h
index 3be7908c90c..e7d7b60b422 100644
--- a/STL_Extension/include/CGAL/Skiplist.h
+++ b/STL_Extension/include/CGAL/Skiplist.h
@@ -75,7 +75,7 @@ public:
       all_iterator
     , typename all_list::iterator
     , T
-    >
+    , std::bidirectional_iterator_tag>
   {
   public:
     all_iterator() {}
@@ -91,7 +91,7 @@ public:
       skip_iterator
     , typename skip_list::iterator
     , T
-    >
+    , std::bidirectional_iterator_tag>
   {
   public:
     skip_iterator() {}