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. 
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 $\gamma$.

References

SHOWING 1-10 OF 18 REFERENCES
A characterization of infinite planar primitive graphs
On the structure of infinite vertex-transitive graphs
A note on fragments of infinite graphs
Results involving automorphisms and fragments of infinite graphs are proved. In particular for a given fragmentC and a vertex-transitive subgroupG of the automorphism group of a connected graph there
Surfaces and Planar Discontinuous Groups
Free groups and graphs.- 2-Dimensional complexes and combinatorial presentations of groups.- Surfaces.- Planar discontinuous groups.- Automorphisms of planar groups.- On the complex analytic theory
A note on bounded automorphisms of infinite graphs
LetX be a connected locally finite graph with vertex-transitive automorphism group. IfX has polynomial growth then the set of all bounded automorphisms of finite order is a locally finite, periodic
Non-Separable and Planar Graphs.
  • H. Whitney
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1931
TLDR
A dual of a graph is defined by combinatorial means, and the paper ends with the theorem that a necessary and sufficient condition that a graph be planar is that it have a dual.
Tree amalgamation of graphs and tessellations of the Cantor sphere
  • B. Mohar
  • Mathematics
    J. Comb. Theory, Ser. B
  • 2006
Locally finite, planar, edge-transitive graphs
What do you do to start reading locally finite planar edge transitive graphs? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has
3-connected planar spaces uniquely embed in the sphere
We characterize those locally connected subsets of the sphere that have a unique embedding in the sphere - i.e., those for which every homeomorphism of the subset to itself extends to a homeomorphism
Planarity and duality of finite and infinite graphs
...
...