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.

  • 1
    $\begingroup$ 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! $\endgroup$
    – L.C. Zhang
    Apr 6, 2021 at 10:19

1 Answer 1


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.