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stackoverflow . com/questions/12886036/deep-copying-a-graph-structure

Not sure about the question. But i will answer some of the specific points in the question.
How to differentiate between different nodes with same data and encountering the same node? A: use the address of the node to differentiate
also how do you make sure you don't run into loops for the exact node? A: as its done in DFS.

Actually this problem can be formed as serialization problem. We need some function to check object identity. We can use Java identityHashCode and put all node of graph to map. Where key if this identityHash.

BFS - for accessing all vertex and edges and handling cycles with love.

for every vertex you take out from queue, clone it. store in hashmap <orinigal,clone>.

if adjacent is found for first time,
clone vertex store in hashmap <orinigal,clone> and add edge in clone adjacency.
else
take out the visited node for hashmap, add edge, put it back.

vertex and edges need to be objects. you can have 2 vertex with same value.