diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/Edge_profile_impl.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/Edge_profile_impl.h
index aa90ad9efff..bcf20379d41 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/Edge_profile_impl.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/Policies/Edge_collapse/Edge_profile_impl.h
@@ -152,7 +152,7 @@ template<class VertexIdxMap
 
   // counterclockwise around v0
   halfedge_descriptor e02 = opposite(prev(v0_v1(),surface_mesh()), surface_mesh());
-  vertex_descriptor v, v2 =target(e02,surface_mesh());
+  vertex_descriptor v=target(e02,surface_mesh()), v2=v;
   while(e02 != endleft) {
     #ifdef CGAL_SMS_EDGE_PROFILE_ALWAYS_NEED_UNIQUE_VERTEX_IN_LINK
     if (vertex_already_inserted.insert(v2).second)
@@ -161,12 +161,14 @@ template<class VertexIdxMap
     bool is_b = is_border(e02);
     e02 = opposite(prev(e02,surface_mesh()), surface_mesh());
     v = target(e02,surface_mesh());
-    if(! is_b){
+    if( !is_b )
       mTriangles.push_back(Triangle(v,v0(),v2) ) ;
-    }
     v2 = v;
   }
-  if(v != vR() && (v!= vertex_descriptor())){
+
+  e02 = opposite(prev(v1_v0(),surface_mesh()), surface_mesh());
+  if(target(e02, surface_mesh())!=v ) // add the vertex if it is not added in the following loop
+  {
     #ifdef CGAL_SMS_EDGE_PROFILE_ALWAYS_NEED_UNIQUE_VERTEX_IN_LINK
     if (vertex_already_inserted.insert(v).second)
     #endif
@@ -174,8 +176,8 @@ template<class VertexIdxMap
   }
 
   // counterclockwise around v1
-  e02 = opposite(prev(v1_v0(),surface_mesh()), surface_mesh());
   v2 = target(e02,surface_mesh());
+  v = v2;
   while(e02 != endright) {
     #ifdef CGAL_SMS_EDGE_PROFILE_ALWAYS_NEED_UNIQUE_VERTEX_IN_LINK
     if (vertex_already_inserted.insert(v2).second)
@@ -184,18 +186,22 @@ template<class VertexIdxMap
     bool is_b = is_border(e02);
     e02 = opposite(prev(e02,surface_mesh()), surface_mesh());
     v = target(e02,surface_mesh());
-    if(! is_b){
+    if( !is_b ){
       mTriangles.push_back(Triangle(v,v1(),v2) ) ;
     }
     v2 = v;
   }
-  if(v != vL() && (v!= vertex_descriptor())){
+
+  if(mLink.empty() || //handle link of an isolated triangle
+     target(opposite(prev(v0_v1(),surface_mesh()), surface_mesh()), surface_mesh())!=v)
+  {
     #ifdef CGAL_SMS_EDGE_PROFILE_ALWAYS_NEED_UNIQUE_VERTEX_IN_LINK
     if (vertex_already_inserted.insert(v).second)
     #endif
     mLink.push_back(v);
   }
-  
+
+  CGAL_assertion(!mLink.empty());
 }
 
 } // namespace Surface_mesh_simplification
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test01.off b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test01.off
new file mode 100644
index 00000000000..821819e7179
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test01.off
@@ -0,0 +1,38 @@
+OFF
+16 18 0
+
+-12 -8 0
+-9 -11 0
+-10 -7 0
+-10 -17 0
+-9 -14 0
+-12 -15 0
+-3 -7 0
+-6 -6 0
+-5 -10 0
+-13 -14 0
+-8 -19 0
+-6 -15 0
+-2 -17 0
+-4 -20 0
+-2 -10 0
+-9 -6 0
+3  0 1 2
+3  3 4 5
+3  6 7 8
+3  5 4 9
+3  9 1 0
+3  10 11 4
+3  12 11 13
+3  10 4 3
+3  7 1 8
+3  8 11 14
+3  2 1 15
+3  11 8 4
+3  11 12 14
+3  14 6 8
+3  4 8 1
+3  10 13 11
+3  9 4 1
+3  15 1 7
+
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test02.off b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test02.off
new file mode 100644
index 00000000000..5ab87105c08
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test02.off
@@ -0,0 +1,37 @@
+OFF
+16 17 0
+
+-12 -8 0
+-9 -11 0
+-10 -7 0
+-10 -17 0
+-9 -14 0
+-12 -15 0
+-3 -7 0
+-6 -6 0
+-5 -10 0
+-13 -14 0
+-8 -19 0
+-6 -15 0
+-2 -17 0
+-4 -20 0
+-2 -10 0
+-9 -6 0
+3  0 1 2
+3  3 4 5
+3  6 7 8
+3  9 1 0
+3  10 11 4
+3  12 11 13
+3  10 4 3
+3  7 1 8
+3  8 11 14
+3  2 1 15
+3  11 8 4
+3  11 12 14
+3  14 6 8
+3  4 8 1
+3  10 13 11
+3  9 4 1
+3  15 1 7
+
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test03.off b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test03.off
new file mode 100644
index 00000000000..75f88a0e899
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test03.off
@@ -0,0 +1,36 @@
+OFF
+16 16 0
+
+-12 -8 0
+-9 -11 0
+-10 -7 0
+-10 -17 0
+-9 -14 0
+-12 -15 0
+-3 -7 0
+-6 -6 0
+-5 -10 0
+-13 -14 0
+-8 -19 0
+-6 -15 0
+-2 -17 0
+-4 -20 0
+-2 -10 0
+-9 -6 0
+3  0 1 2
+3  3 4 5
+3  6 7 8
+3  9 1 0
+3  10 11 4
+3  12 11 13
+3  10 4 3
+3  7 1 8
+3  8 11 14
+3  2 1 15
+3  11 8 4
+3  11 12 14
+3  14 6 8
+3  4 8 1
+3  10 13 11
+3  15 1 7
+
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test04.off b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test04.off
new file mode 100644
index 00000000000..58526fa9ee1
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test04.off
@@ -0,0 +1,31 @@
+OFF
+14 13 0
+
+-12 -8 0
+-9 -11 0
+-10 -7 0
+-9 -14 0
+-3 -7 0
+-6 -6 0
+-5 -10 0
+-13 -14 0
+-8 -19 0
+-6 -15 0
+-2 -17 0
+-4 -20 0
+-2 -10 0
+-9 -6 0
+3  0 1 2
+3  4 5 6
+3  7 1 0
+3  10 9 11
+3  5 1 6
+3  6 9 12
+3  2 1 13
+3  9 6 3
+3  9 10 12
+3  12 4 6
+3  3 6 1
+3  8 11 9
+3  13 1 5
+
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test05.off b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test05.off
new file mode 100644
index 00000000000..7a7517d60b3
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test05.off
@@ -0,0 +1,36 @@
+OFF
+16 16 0
+
+-12 -8 0
+-9 -11 0
+-10 -7 0
+-10 -17 0
+-9 -14 0
+-12 -15 0
+-3 -7 0
+-6 -6 0
+-5 -10 0
+-13 -14 0
+-8 -19 0
+-6 -15 0
+-2 -17 0
+-4 -20 0
+-2 -10 0
+-9 -6 0
+3  0 1 2
+3  3 4 5
+3  6 7 8
+3  9 1 0
+3  10 11 4
+3  12 11 13
+3  10 4 3
+3  7 1 8
+3  8 11 14
+3  2 1 15
+3  11 8 4
+3  11 12 14
+3  4 8 1
+3  10 13 11
+3  9 4 1
+3  15 1 7
+
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test06.off b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test06.off
new file mode 100644
index 00000000000..cc30830bc25
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test06.off
@@ -0,0 +1,34 @@
+OFF
+16 14 0
+
+-12 -8 0
+-9 -11 0
+-10 -7 0
+-10 -17 0
+-9 -14 0
+-12 -15 0
+-3 -7 0
+-6 -6 0
+-5 -10 0
+-13 -14 0
+-8 -19 0
+-6 -15 0
+-2 -17 0
+-4 -20 0
+-2 -10 0
+-9 -6 0
+3  0 1 2
+3  3 4 5
+3  6 7 8
+3  9 1 0
+3  10 11 4
+3  12 11 13
+3  10 4 3
+3  7 1 8
+3  2 1 15
+3  11 8 4
+3  11 12 14
+3  4 8 1
+3  10 13 11
+3  15 1 7
+
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test07.off b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test07.off
new file mode 100644
index 00000000000..de566861221
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test07.off
@@ -0,0 +1,23 @@
+OFF
+10 9 0
+
+-9 -11 0
+-10 -17 0
+-9 -14 0
+-12 -15 0
+-5 -10 0
+-8 -19 0
+-6 -15 0
+-2 -17 0
+-4 -20 0
+-2 -10 0
+3  1 2 3
+3  5 6 2
+3  7 6 8
+3  5 2 1
+3  4 6 9
+3  6 4 2
+3  6 7 9
+3  2 4 0
+3  5 8 6
+
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test08.off b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test08.off
new file mode 100644
index 00000000000..3637437a8ee
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test08.off
@@ -0,0 +1,28 @@
+OFF
+13 11 0
+
+-12 -8 0
+-9 -11 0
+-10 -7 0
+-9 -14 0
+-6 -6 0
+-5 -10 0
+-13 -14 0
+-8 -19 0
+-6 -15 0
+-2 -17 0
+-4 -20 0
+-2 -10 0
+-9 -6 0
+3  0 1 2
+3  6 1 0
+3  7 8 3
+3  9 8 10
+3  4 1 5
+3  2 1 12
+3  8 5 3
+3  8 9 11
+3  3 5 1
+3  7 10 8
+3  12 1 4
+
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test09.off b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test09.off
new file mode 100644
index 00000000000..6c25a242cd1
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test09.off
@@ -0,0 +1,10 @@
+OFF
+4 2 0
+
+-9 -11 0
+-9 -14 0
+-5 -10 0
+-6 -15 0
+3  3 2 1
+3  1 2 0
+
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test10.off b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test10.off
new file mode 100644
index 00000000000..eb03f5b82d7
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/data_link/test10.off
@@ -0,0 +1,8 @@
+OFF
+3 1 0
+
+-9 -11 0
+-9 -14 0
+-5 -10 0
+3  1 2 0
+
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/test_edge_profile_link.cmd b/Surface_mesh_simplification/test/Surface_mesh_simplification/test_edge_profile_link.cmd
new file mode 100644
index 00000000000..8b8d3c31aa1
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/test_edge_profile_link.cmd
@@ -0,0 +1,11 @@
+data_link/test01.off
+data_link/test02.off
+data_link/test03.off
+data_link/test04.off
+data_link/test05.off
+data_link/test06.off
+data_link/test07.off
+data_link/test08.off
+data_link/test09.off
+data_link/test10.off
+data/elephant.off
diff --git a/Surface_mesh_simplification/test/Surface_mesh_simplification/test_edge_profile_link.cpp b/Surface_mesh_simplification/test/Surface_mesh_simplification/test_edge_profile_link.cpp
new file mode 100644
index 00000000000..32770928602
--- /dev/null
+++ b/Surface_mesh_simplification/test/Surface_mesh_simplification/test_edge_profile_link.cpp
@@ -0,0 +1,122 @@
+#include <CGAL/Simple_cartesian.h>
+#include <CGAL/Surface_mesh.h>
+#include <CGAL/Surface_mesh_simplification/Policies/Edge_collapse/Edge_profile.h>
+#include <boost/tuple/tuple.hpp>
+#include <boost/tuple/tuple_comparison.hpp>
+#include <fstream>
+
+typedef CGAL::Simple_cartesian<double> K;
+typedef CGAL::Surface_mesh<K::Point_3> Mesh;
+typedef CGAL::Surface_mesh_simplification::Edge_profile<Mesh> Profile;
+
+void naive_all_triangles(Mesh::Halfedge_index h, Mesh& m, std::set<Mesh::Face_index>& triangles)
+{
+  BOOST_FOREACH(Mesh::Halfedge_index hh, CGAL::halfedges_around_source(h, m))
+  {
+    if (!is_border(hh, m))
+      triangles.insert(face(hh,m));
+  }
+  BOOST_FOREACH(Mesh::Halfedge_index hh, CGAL::halfedges_around_target(h, m))
+  {
+    if (!is_border(hh, m))
+      triangles.insert(face(hh,m));
+  }
+}
+
+void naive_link_vertices(Mesh::Halfedge_index h, Mesh& m,
+                         const std::set<Mesh::Face_index>& triangles,
+                         std::set<Mesh::Vertex_index>& link_vertices)
+{
+  BOOST_FOREACH(Mesh::Face_index f, triangles)
+  {
+    BOOST_FOREACH(Mesh::Halfedge_index h, CGAL::halfedges_around_face(halfedge(f, m), m))
+    {
+      link_vertices.insert(target(h, m));
+    }
+  }
+  link_vertices.erase( source(h, m) );
+  link_vertices.erase( target(h, m) );
+}
+
+struct A{};
+
+boost::tuple<Mesh::Vertex_index, Mesh::Vertex_index, Mesh::Vertex_index>
+make_canonical_tuple(Mesh::Vertex_index v1, Mesh::Vertex_index v2, Mesh::Vertex_index v3)
+{
+  Mesh::Vertex_index vs[3]={v1, v2, v3};
+  std::sort(&vs[0], &vs[0]+3);
+  return boost::make_tuple(vs[0], vs[1], vs[2]);
+}
+
+boost::tuple<Mesh::Vertex_index, Mesh::Vertex_index, Mesh::Vertex_index>
+make_canonical_tuple(const Profile::Triangle& t)
+{
+  return make_canonical_tuple(t.v0, t.v1, t.v2);
+}
+
+boost::tuple<Mesh::Vertex_index, Mesh::Vertex_index, Mesh::Vertex_index>
+make_canonical_tuple(Mesh::Face_index f, Mesh& m)
+{
+  Mesh::Halfedge_index h=halfedge(f, m);
+  return make_canonical_tuple(source(h,m), target(h,m), target(next(h, m), m));
+}
+
+void test(const char* fname)
+{
+  Mesh m;
+  std::ifstream input(fname);
+  assert( bool(input) );
+  input >> m;
+  assert(num_vertices(m)!=0);
+  A a;
+
+  BOOST_FOREACH(Mesh::Halfedge_index h, halfedges(m))
+  {
+    std::set<Mesh::Face_index> triangles;
+    naive_all_triangles(h, m, triangles);
+
+    Profile profile(h, m, a, get(boost::vertex_point, m), a, true);
+
+    if (CGAL::Euler::does_satisfy_link_condition(edge(h, m), m))
+    {
+      std::set<Mesh::Vertex_index> link_vertices;
+      naive_link_vertices(h, m, triangles, link_vertices);
+      assert( link_vertices.size()==profile.link().size() );
+      assert( std::set<Mesh::Vertex_index>(profile.link().begin(),
+                                           profile.link().end()).size()
+              == link_vertices.size() );
+
+      BOOST_FOREACH(const Mesh::Vertex_index& v, profile.link())
+      {
+        assert( link_vertices.count(v) == 1 );
+      }
+    }
+
+    assert(triangles.size() == profile.triangles().size());
+    std::set<boost::tuple<Mesh::Vertex_index, Mesh::Vertex_index, Mesh::Vertex_index> > triple_set;
+    BOOST_FOREACH(const Profile::Triangle& t, profile.triangles())
+    {
+      triple_set.insert( make_canonical_tuple(t) );
+    }
+    BOOST_FOREACH(Mesh::Face_index f, triangles)
+    {
+      assert( triple_set.count( make_canonical_tuple(f, m) ) == 1 );
+    }
+  }
+}
+
+int main(int argc, char** argv)
+{
+  for (int i=1; i<argc; ++i)
+  {
+    std::cout << "Testing " << argv[i] << "\n";
+    test(argv[i]);
+  }
+  if (argc==1)
+  {
+    std::cout << "No file provided, nothing tested\n";
+  }
+
+  return 0;
+}
+