<p>This is a question about the correct reference for a result that seems to appear frequently in the literature on planar graph isomorphism. In <a href="http://dl.acm.org/citation.cfm?id=1740064" rel="noreferrer">"A $V \log V$ Algorithm for Isomorphism of Triconnected Planar Graphs"</a>, 
Hopcraft and Tarjan mention that a 3-connected planar graph has an essentially unique embedding into the plane (up to mirror symmetry), and attribute this result to Whitney (1933) <a href="https://www.jstor.org/stable/2371125" rel="noreferrer">"A set of topological invariants for graphs"</a>. I've noticed this citation reappearing in various places (for example <a href="http://dl.acm.org/citation.cfm?id=804670" rel="noreferrer">here</a>, <a href="https://www.cse.iitb.ac.in/~nutan/papers/planarGI-ccc.pdf" rel="noreferrer">here</a>, and <a href="https://arxiv.org/pdf/1310.1896.pdf" rel="noreferrer">here</a>), but I couldn't find the result in Whitney's article itself -- am I just overlooking something? The paper is dense, and builds on a series of other articles he published in succession... Still, this result sounds a lot like Theorem 11 of Whitney (1932) <a href="http://www.jstor.org/stable/2371086" rel="noreferrer">"Congruent Graphs and the Connectivity of Graphs"</a>, which states that a 3-connected planar graph has a unique dual. Was that the intended reference?</p>

CC BY-SA 4.0
2023-06-09T16:13:21.350Z
Years later, I'm having the same issue. Babai (Handbook of Combinatorics, Chapter 27, "Automorphism Groups, Isomorphism, Reconstruction") also refers to Whitney (1932) "Congruent Graphs and the Connectivity of Graphs", but I also couldn't find it in there. Another [arXiv paper](https://arxiv.org/abs/2005.04895) I found refers to Whitney (1993) "2-Isomorphic graphs", but again I couldn't find the result in that paper. Maybe the point is that the result follows from Whitney's work, but was never stated explicitly as a theorem in his papers? Have you since been able to find the right reference?
J. van Dobben de Bruyn
CC BY-SA 4.0
2023-06-09T20:23:45.420Z
It's been a while since I looked at this, and I don't remember! (I do remember having some more non-conclusive followup discussion over on the [nlab forum](https://nforum.ncatlab.org/discussion/7955/whitneys-theorem-on-uniqueness-of-embeddings-of-3connected-graphs-in-the-plane/) shortly afterwards.). But looking again at Whitney's [2-isomorphic graphs](https://www.jstor.org/stable/pdf/2371127.pdf), I do think it is a fair reference -- the first corollary on the second page seems to state a close variant of this result in a slightly unfamiliar language.
Noam Zeilberger
CC BY-SA 4.0
2023-06-09T20:28:01.790Z
I also had a look in Tutte's _Graph Theory_, and he gives a more recognizable statement (and proof) of this result as Theorem XI.52: "Let G be a 3-connected planar graph, not a vertex-graph, a link-graph, or a circuit. Then G has one and only one planar mesh". And he says by another result in the book, this can be identified with "Whitney's Theorem that a 3-connected planar graph can be drawn in the plane [...] in essentially only one way", citing Whitney's "2-isomorphic graphs" and the predecessor paper "Nonseparable and planar graphs".
Noam Zeilberger
<reference-request><graph-theory><planar-graphs><ho.history-overview>
reference clarification: Whitney's theorem on unique embeddability of 3-connected planar graphs?
Noam Zeilberger
# reference clarification: Whitney's theorem on unique embeddability of 3-connected planar graphs?

(Asked by: username_0 on 2017-07-02)

This is a question about the correct reference for a result that seems to appear frequently in the literature on planar graph isomorphism. In ["A $V \log V$ Algorithm for Isomorphism of Triconnected Planar Graphs"](http://dl.acm.org/citation.cfm?id=1740064),
Hopcraft and Tarjan mention that a 3-connected planar graph has an essentially unique embedding into the plane (up to mirror symmetry), and attribute this result to Whitney (1933) ["A set of topological invariants for graphs"](https://www.jstor.org/stable/2371125). I've noticed this citation reappearing in various places (for example [here](http://dl.acm.org/citation.cfm?id=804670), [here](https://www.cse.iitb.ac.in/~nutan/papers/planarGI-ccc.pdf), and [here](https://arxiv.org/pdf/1310.1896.pdf)), but I couldn't find the result in Whitney's article itself -- am I just overlooking something? The paper is dense, and builds on a series of other articles he published in succession... Still, this result sounds a lot like Theorem 11 of Whitney (1932) ["Congruent Graphs and the Connectivity of Graphs"](http://www.jstor.org/stable/2371086), which states that a 3-connected planar graph has a unique dual. Was that the intended reference?

**Tags:** reference-request, graph-theory, planar-graphs, ho.history-overview

> **Comments:**
> **username_1**: Years later, I'm having the same issue. Babai (Handbook of Combinatorics, Chapter 27, "Automorphism Groups, Isomorphism, Reconstruction") also refers to Whitney (1932) "Congruent Graphs and the Connectivity of Graphs", but I also couldn't find it in there. Another [arXiv paper](https://arxiv.org/abs/2005.04895) I found refers to Whitney (1993) "2-Isomorphic graphs", but again I couldn't find the result in that paper. Maybe the point is that the result follows from Whitney's work, but was never stated explicitly as a theorem in his papers? Have you since been able to find the right reference?  
> **username_0**: It's been a while since I looked at this, and I don't remember! (I do remember having some more non-conclusive followup discussion over on the [nlab forum](https://nforum.ncatlab.org/discussion/7955/whitneys-theorem-on-uniqueness-of-embeddings-of-3connected-graphs-in-the-plane/) shortly afterwards.). But looking again at Whitney's [2-isomorphic graphs](https://www.jstor.org/stable/pdf/2371127.pdf), I do think it is a fair reference -- the first corollary on the second page seems to state a close variant of this result in a slightly unfamiliar language.  
> **username_0**: I also had a look in Tutte's _Graph Theory_, and he gives a more recognizable statement (and proof) of this result as Theorem XI.52: "Let G be a 3-connected planar graph, not a vertex-graph, a link-graph, or a circuit. Then G has one and only one planar mesh". And he says by another result in the book, this can be identified with "Whitney's Theorem that a 3-connected planar graph can be drawn in the plane [...] in essentially only one way", citing Whitney's "2-isomorphic graphs" and the predecessor paper "Nonseparable and planar graphs".  

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## 0 answers

