diff --git a/Constrained_triangulation_3/test/Constrained_triangulation_3/CMakeLists.txt b/Constrained_triangulation_3/test/Constrained_triangulation_3/CMakeLists.txt
index 1465facf854..9ed7e0f55e7 100644
--- a/Constrained_triangulation_3/test/Constrained_triangulation_3/CMakeLists.txt
+++ b/Constrained_triangulation_3/test/Constrained_triangulation_3/CMakeLists.txt
@@ -98,6 +98,7 @@ CGAL_add_cdt3_from_off_test("mpi")
 CGAL_add_cdt3_from_off_test("3torus")
 CGAL_add_cdt3_from_off_test("cheese-selection")
 CGAL_add_cdt3_from_off_test("cheese-selection-2")
+CGAL_add_cdt3_from_off_test("non_manifold_face_graph")
 
 function(CGAL_add_cdt3_from_local_off_test data_name)
   CGAL_add_cdt3_from_off_test_aux(${data_name} ${CMAKE_CURRENT_SOURCE_DIR}/data)
diff --git a/Data/data/meshes/non_manifold_face_graph.off b/Data/data/meshes/non_manifold_face_graph.off
new file mode 100644
index 00000000000..b18f4729134
--- /dev/null
+++ b/Data/data/meshes/non_manifold_face_graph.off
@@ -0,0 +1,39 @@
+OFF
+24 11 0
+
+1.5 0.5 1
+0.5 0.5 0
+0.5 0.5 1
+0 0 0
+0 0.5 0
+0.5 0.5 0
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+0.5 0 0
+1 0 0
+1 0.5 0
+1 1 0
+1.5 1.5 1
+2.5 1.5 1
+2.5 0.5 1
+-0.5 -0.5 1
+2.5 -0.5 1
+0.5 1.5 1
+-0.5 0.5 1
+1.5 0.5 1
+1.5 -0.5 1
+-0.5 1.5 1
+0.5 0.5 1
+0.5 -0.5 1
+3  2 1 0
+4  4 3 8 5
+4  7 4 5 6
+4  5 8 9 10
+4  6 5 10 11
+4  12 19 14 13
+4  19 20 16 14
+4  17 22 19 12
+4  22 23 20 19
+4  21 18 22 17
+4  18 15 23 22
+