// the shape inference is complete.
  BackEdgeHelper back_edge;
  TF_RETURN_IF_ERROR(back_edge.Remove(graph));
  TF_RETURN_IF_ERROR(PropagateShapes(graph, arg_shapes,
                                     back_edge.RemovedEdges(), &shape_refiner));
  TF_RETURN_IF_ERROR(back_edge.Replace());

  // Currently information does not flow "backward" from consumers to producers

  TF_RETURN_IF_ERROR(back_edge_helper_.Remove(graph_.get()));
  VLOG(1) << "Found " << (back_edge_helper_.RemovedEdges().size())
          << " backedges.";

  // Creates a map for quickly identifying whether a node output is a backedge.
  for (const auto& edge : back_edge_helper_.RemovedEdges()) {
    if (back_edge_node_output_.find(edge.src) != back_edge_node_output_.end() &&