What is good practice? A copy constructor or a defensive copy method

What is good practice? A copy constructor or a defensive copy method

I have a class Graph and I need a copy of that graph. I am going to modify
the internals of graph object ( eg: delete an edge etc. ). I have 2 ways
to implement a graph.
A copy constructor
A method called 'getGraph() { return new Graph(this)}'. This method
getGraph can do a defensive copy.
The only advantage of copy constructor, from my understanding is
copy-at-will. This means if I dont want to modify graph object, there is
not need for 'getGraph' to do a defensive copy.
Now coming back to my question.
Is it better to use a copy constructor or is it better to use a function
which returns copy of self object ?
Why ?