Connected but not path-connected subspaces of infinite graphs

@article{Georgakopoulos2007ConnectedBN,
  title={Connected but not path-connected subspaces of infinite graphs},
  author={Agelos Georgakopoulos},
  journal={Combinatorica},
  year={2007},
  volume={27},
  pages={683-698}
}
Solving a problem of Diestel [9] relevant to the theory of cycle spaces of infinite graphs, we show that the Freudenthal compactification of a locally finite graph can have connected subsets that are not path-connected. However we prove that connectedness and path-connectedness to coincide for all but a few sets, which have a complicated structure. 

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References

SHOWING 1-10 OF 30 REFERENCES

On end degrees and infinite cycles in locally finite graphs

It is proved that the edge set of a locally finite graph G lies in C(G) if and only if every vertex and every end has even degree.

The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits

  • H. Bruhn
  • Mathematics
    J. Comb. Theory, Ser. B
  • 2004

On Infinite Cycles II

The spanning trees whose fundamental cycles generate this cycle space are characterized, and infinite analogues to the standard characterizations of finite cycle spaces in terms of edge-decomposition into single cycles and orthogonality to cuts are proved.

Topological paths, cycles and spanning trees in infinite graphs

On Infinite Cycles I

We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S1 in the graph compactified by its ends. We prove

Martin and end compactifications for non-locally finite graphs

We consider a connected graph, having countably infinite vertex set X, which is permitted to have vertices of infinite degree. For a transient irreducible transition matrix P corresponding to a

Duality in Infinite Graphs

This work shows that obstructions fall away when duality is reinterpreted on the basis of a ‘singular’ approach to graph homology, whose cycles are defined topologically in a space formed by the graph together with its ends and can be infinite.

Graph-theoretical versus topological ends of graphs

The Cycle Space of an Infinite Graph

  • R. Diestel
  • Mathematics
    Combinatorics, Probability and Computing
  • 2005
A new ‘singular’ approach is presented that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of $S^1$ in the space formed by the graph together with its ends.

Forcing highly connected subgraphs

Mader’s theorem does not extend to infinite graphs, not even to locally finite ones, but a lower bound of at least 2k for the edge-degree at the ends and the degree at the vertices does suffice to ensure the existence of (k+1)-edge-connected subgraphs in arbitrary graphs.