Connected but not path-connected subspaces of infinite graphs

Georgakopoulos, Angelos

doi:10.1007/s00493-007-2188-6

Cited by 8 publications

(12 citation statements)

References 18 publications

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“…Closed subspaces of the Freudenthal compactification are compact and inherit therefore the nice properties of the Freudenthal compactification itself. But, the general subspaces do not (as shown in [16]). Nevertheless they satisfy Menger's Theorem as we show.…”

Section: Introductionmentioning

confidence: 88%

Graph-like continua, augmenting arcs, and Menger’s theorem

Thomassen

Vella

2008

Combinatorica

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We show that an adaptation of the augmenting path method for graphs proves Menger's Theorem for wide classes of topological spaces. For example, it holds for locally compact, locally connected, metric spaces, as already known. The method lends itself particularly well to another class of spaces, namely the locally arcwise connected, hereditarily locally connected, metric spaces. Finally, it applies to every space where every point can be separated from every closed set not containing it by a finite set, in particular to every subspace of the Freudenthal compactification of a locally finite, connected graph. While closed subsets of such a space behave nicely in that they are compact and locally connected (and therefore locally arcwise connected), the general subspaces do not: They may be connected without being arcwise connected. Nevertheless, they satisfy Menger's Theorem.

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Section: Introductionmentioning

confidence: 88%

Graph-like continua, augmenting arcs, and Menger’s theorem

Thomassen

Vella

2008

Combinatorica

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“…A continuous map from the real unit interval [0, 1] to a topological space X is a (topological) path in X. The following lemma can be found in [9]. It will be used in Section 4.…”

Section: • |E(g) ∩ F | Is Even For Every Finite Cut F Of Gmentioning

confidence: 99%

Infinite Hamilton cycles in squares of locally finite graphs

Georgakopoulos¹

2009

Advances in Mathematics

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“…Finally, we prove the main result of this section, namely, G C is 2-arc-connected. This follows from a theorem by Georgakopoulos [7] concerning connected but not pathconnected subspaces of locally finite graphs. Note that since |G| is Hausdorff, pathconnectedness is equivalent to arc-connectedness.…”

Section: Lemma 12mentioning

confidence: 94%

“…Theorem 14 (Georgakopoulos [7]). Given any locally finite connected graph G, a connected subspace X of |G| is path-connected unless it satisfies the following assertions:…”

Section: Lemma 12mentioning

confidence: 99%

Contractible Edges in 2-Connected Locally Finite Graphs

Chan¹

2015

Electron. J. Combin.

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In this paper, we prove that every contraction-critical 2-connected infinite graph has no vertex of finite degree and contains uncountably many ends. Then, by investigating the distribution of contractible edges in a 2-connected locally finite infinite graph $G$, we show that the closure of the subgraph induced by all the contractible edges in the Freudenthal compactification of $G$ is 2-arc-connected. Finally, we characterize all 2-connected locally finite outerplanar graphs nonisomorphic to $K_3$ as precisely those graphs such that every vertex is incident to exactly two contractible edges as well as those graphs such that every finite bond contains exactly two contractible edges.

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Connected but not path-connected subspaces of infinite graphs

Abstract: Solving a problem of Diestel [8] 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 pathconnectedness do coincide for all but a few sets, which have a complicated structure.

Cited by 8 publications

References 18 publications

Graph-like continua, augmenting arcs, and Menger’s theorem

Graph-like continua, augmenting arcs, and Menger’s theorem

Infinite Hamilton cycles in squares of locally finite graphs

Contractible Edges in 2-Connected Locally Finite Graphs

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