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A planar graph is one which has a plane embedding. Two drawings are topologically isomorphic if one can be continuously deformed into the other. If we wrap a drawing onto a sphere, and then off again, we can move any face to be the exterior face.

We know the following theorem.

Theorem (Whitney) If $G$ is 3-connected, any two planar embeddings are equivalent.

My question is as the title says.

Is graph's planar embedding unique if each block of one planar graph is 3-connected?

Connectivity of the graph may even be one. I don’t know if there are any counterexamples or it may be true. Not sure if this is a proven fact.

PS: A block of a graph G is a maximal subgraph which is either an isolated vertex, a bridge edge, or a 2-connected subgraph.

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    $\begingroup$ No, in general you can flip blocks over. $\endgroup$ Apr 6, 2021 at 10:11
  • $\begingroup$ @BrendanMcKay Thank you for your answer. I feel you are right, but I still have a doubt. I don't quite understand why is it different after flipping? Could you elaborate on the reason? Thank you! $\endgroup$
    – lcz
    Apr 6, 2021 at 10:19

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Thanks for Brendan McKay's help. The following two are indeed not same embedding. When the subgraph on the right is reversed to the 3 face of the graph on the left subgraph. enter image description here

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