How to add edges while keeping the original graph unchanged

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I see that the commands for adding vertices and edges in Sage will alter the original graph. So I write the following code which makes the origin graph unchanged.

def addedge(g, edges_to_add):
    G1 = g.copy() 
    G1.add_edge(edges_to_add)  
    return G1

I need to sequentially add one edge from the complement of the graph to observe their properties, such as Hamiltonicity.

 def addedge(g, edges_to_add):
        G1 = g.copy() 
        G1.add_edge(edges_to_add)  
        return G1
s = 'J~f[MN@PQg?'
G = Graph(s, sparse=True); 
complement_edges = G.complement().edges(labels=False, sort=False)

for edge in complement_edges:
    G1=addedge(G,edge)
    if not G1.is_hamiltonian():
        print("The modified graph is non-Hamiltonian  with the added edge:", edge)
    else:
        print()
    #print("The modified graph is Hamiltonian with the added edge:",edge)

I am not sure if this edge addition is highly efficient because, as we discussed in previous issues (see common in the answer )(calculate-the-toughness-of-a-graph, it's best to avoid copy() if it's being repeatedly used.