Imprimitivity of locally finite, 1-ended, planar graphs

@article{Sirn2012ImprimitivityOL,
  title={Imprimitivity of locally finite, 1-ended, planar graphs},
  author={Jozef Sir{\'a}n and Mark E. Watkins},
  journal={Ars Math. Contemp.},
  year={2012},
  volume={5},
  pages={217-221}
}
Using results from group theory, we offer a concise proof of the imprimitivity of locally finite, vertex-transitive, 1-ended planar graphs, a result previously established by J. E. Graver and M. E. Watkins (2004) using graph-theoretical methods. 
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Lobe, edge, and arc transitivity of graphs of connectivity 1
TLDR
It is shown that, given any biconnected graph $\Lambda$ and a "code" assigned to each orbit of Aut, there exists a unique lobe-transitive graph $\Gamma$ of connectivity 1 whose lobes are copies of $\Lamba$ and is consistent with the given code at every vertex of $\gamma$.

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