code: def min_fill_heuristic(G):
      
      # we need only deal with the adjacency structure of G. We will also
      # be manipulating it directly so let's go ahead and make a new one
      adj = {v: set(G[v]) for v in G}
  
      num_nodes = len(adj)
  
      # preallocate the return values
      order = [0] * num_nodes
      upper_bound = 0
  
      for i in range(num_nodes):
          # get the node that adds the fewest number of edges when eliminated from the graph
          v = min(adj, key=lambda x: _min_fill_needed_edges(adj, x))
  
          # if the number of neighbours of v is higher than upper_bound, update
          dv = len(adj[v])
          if dv > upper_bound:
              upper_bound = dv
  
          # make v simplicial by making its neighborhood a clique then remove the
          # node
          _elim_adj(adj, v)
          order[i] = v
  
      return upper_bound, order
docs: Computes an upper bound on the treewidth of graph G based on
      the min-fill heuristic for the elimination ordering.
  
      Parameters
      ----------
      G : NetworkX graph
          The graph on which to compute an upper bound for the treewidth.
  
      Returns
      -------
      treewidth_upper_bound : int
          An upper bound on the treewidth of the graph G.
  
      order : list
          An elimination order that induces the treewidth.
  
      Examples
      --------
      This example computes an upper bound for the treewidth of the :math:`K_4`
      complete graph.
  
      >>> import dwave_networkx as dnx
      >>> import networkx as nx
      >>> K_4 = nx.complete_graph(4)
      >>> tw, order = dnx.min_fill_heuristic(K_4)
  
      References
      ----------
      Based on the algorithm presented in [GD]_