References analysis of the Wikipedia article "End (graph theory)" in the English version

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However, as Krön & Möller (2008) point out, ends of graphs were already considered by Freudenthal (1945). Krön, Bernhard; Möller, Rögnvaldur G. (2008), "Metric ends, fibers and automorphisms of graphs" (PDF), Mathematische Nachrichten, 281 (1): 62–74, doi:10.1002/mana.200510587, MR 2376468, S2CID 18378856 Freudenthal, Hans (1945), "Über die Enden diskreter Räume und Gruppen", Commentarii Mathematici Helvetici, 17: 1–38, doi:10.1007/bf02566233, MR 0012214, S2CID 125014683

Halin (1964). Halin, Rudolf (1964), "Über unendliche Wege in Graphen", Mathematische Annalen, 157 (2): 125–137, doi:10.1007/bf01362670, hdl:10338.dmlcz/102294, MR 0170340, S2CID 122125458

E.g., this is the form of the equivalence relation used by Diestel & Kühn (2003). Diestel, Reinhard; Kühn, Daniela (2003), "Graph-theoretical versus topological ends of graphs", Journal of Combinatorial Theory, Series B, 87 (1): 197–206, doi:10.1016/S0095-8956(02)00034-5, MR 1967888

The haven nomenclature, and the fact that two rays define the same haven if and only if they are equivalent, is due to Robertson, Seymour & Thomas (1991). Diestel & Kühn (2003) proved that every haven comes from an end, completing the bijection between ends and havens, using a different nomenclature in which they called havens "directions". Robertson, Neil; Seymour, Paul; Thomas, Robin (1991), "Excluding infinite minors", Discrete Mathematics, 95 (1–3): 303–319, doi:10.1016/0012-365X(91)90343-Z, MR 1141945 Diestel, Reinhard; Kühn, Daniela (2003), "Graph-theoretical versus topological ends of graphs", Journal of Combinatorial Theory, Series B, 87 (1): 197–206, doi:10.1016/S0095-8956(02)00034-5, MR 1967888

The proof by Diestel & Kühn (2003) that every haven can be defined by a ray is nontrivial and involves two cases. If the set

S=\bigcap _{X}\left(\beta (X)\cup X\right)

(where

X

ranges over all finite sets of vertices) is infinite, then there exists a ray that passes through infinitely many vertices of

S

, which necessarily determines

\beta

. On the other hand, if

S

is finite, then Diestel & Kühn (2003) show that in this case there exists a sequence of finite sets

X_{i}

that separate the end from all points whose distance from an arbitrarily chosen starting point in

G\setminus S

i

. In this case, the haven is defined by any ray that is followed by a robber using the haven to escape police who land at set

X_{i}

in round

i

of the pursuit–evasion game. Diestel, Reinhard; Kühn, Daniela (2003), "Graph-theoretical versus topological ends of graphs", Journal of Combinatorial Theory, Series B, 87 (1): 197–206, doi:10.1016/S0095-8956(02)00034-5, MR 1967888 Diestel, Reinhard; Kühn, Daniela (2003), "Graph-theoretical versus topological ends of graphs", Journal of Combinatorial Theory, Series B, 87 (1): 197–206, doi:10.1016/S0095-8956(02)00034-5, MR 1967888

More precisely, in the original formulation of this result by Halin (1964) in which ends are defined as equivalence classes of rays, every equivalence class of rays of

G

contains a unique nonempty equivalence class of rays of the spanning forest. In terms of havens, there is a one-to-one correspondence of havens of order

\aleph _{0}

between

G

and its spanning tree

T

for which

\beta _{T}(X)\subset \beta _{G}(X)

for every finite set

X

and every corresponding pair of havens

\beta _{T}

and

\beta _{G}

. Halin, Rudolf (1964), "Über unendliche Wege in Graphen", Mathematische Annalen, 157 (2): 125–137, doi:10.1007/bf01362670, hdl:10338.dmlcz/102294, MR 0170340, S2CID 122125458

Seymour & Thomas (1991); Thomassen (1992); Diestel (1992). Seymour, Paul; Thomas, Robin (1991), "An end-faithful spanning tree counterexample", Proceedings of the American Mathematical Society, 113 (4): 1163–1171, doi:10.2307/2048796, JSTOR 2048796, MR 1045600 Thomassen, Carsten (1992), "Infinite connected graphs with no end-preserving spanning trees", Journal of Combinatorial Theory, Series B, 54 (2): 322–324, doi:10.1016/0095-8956(92)90059-7, hdl:10338.dmlcz/127625, MR 1152455

Diestel & Kühn (2003). Diestel, Reinhard; Kühn, Daniela (2003), "Graph-theoretical versus topological ends of graphs", Journal of Combinatorial Theory, Series B, 87 (1): 197–206, doi:10.1016/S0095-8956(02)00034-5, MR 1967888

Diestel (2006). Diestel, Reinhard (2006), "End spaces and spanning trees", Journal of Combinatorial Theory, Series B, 96 (6): 846–854, doi:10.1016/j.jctb.2006.02.010, MR 2274079

Halin (1965); Diestel (2004). Halin, Rudolf (1965), "Über die Maximalzahl fremder unendlicher Wege in Graphen", Mathematische Nachrichten, 30 (1–2): 63–85, doi:10.1002/mana.19650300106, MR 0190031 Diestel, Reinhard (2004), "A short proof of Halin's grid theorem", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 74: 237–242, doi:10.1007/BF02941538, MR 2112834, S2CID 124603912

Stallings (1968, 1971). Stallings, John R. (1968), "On torsion-free groups with infinitely many ends", Annals of Mathematics, Second Series, 88 (2): 312–334, doi:10.2307/1970577, JSTOR 1970577, MR 0228573 Stallings, John R. (1971), Group theory and three-dimensional manifolds: A James K. Whittemore Lecture in Mathematics given at Yale University, 1969, Yale Mathematical Monographs, vol. 4, New Haven, Conn.: Yale University Press, MR 0415622

doi.org

Halin (1964). Halin, Rudolf (1964), "Über unendliche Wege in Graphen", Mathematische Annalen, 157 (2): 125–137, doi:10.1007/bf01362670, hdl:10338.dmlcz/102294, MR 0170340, S2CID 122125458

The proof by Diestel & Kühn (2003) that every haven can be defined by a ray is nontrivial and involves two cases. If the set

S=\bigcap _{X}\left(\beta (X)\cup X\right)

(where

X

ranges over all finite sets of vertices) is infinite, then there exists a ray that passes through infinitely many vertices of

S

, which necessarily determines

\beta

. On the other hand, if

S

is finite, then Diestel & Kühn (2003) show that in this case there exists a sequence of finite sets

X_{i}

that separate the end from all points whose distance from an arbitrarily chosen starting point in

G\setminus S

i

. In this case, the haven is defined by any ray that is followed by a robber using the haven to escape police who land at set

X_{i}

in round

i

More precisely, in the original formulation of this result by Halin (1964) in which ends are defined as equivalence classes of rays, every equivalence class of rays of

G

contains a unique nonempty equivalence class of rays of the spanning forest. In terms of havens, there is a one-to-one correspondence of havens of order

\aleph _{0}

between

G

and its spanning tree

T

for which

\beta _{T}(X)\subset \beta _{G}(X)

for every finite set

X

and every corresponding pair of havens

\beta _{T}

and

\beta _{G}

. Halin, Rudolf (1964), "Über unendliche Wege in Graphen", Mathematische Annalen, 157 (2): 125–137, doi:10.1007/bf01362670, hdl:10338.dmlcz/102294, MR 0170340, S2CID 122125458

Diestel (2006). Diestel, Reinhard (2006), "End spaces and spanning trees", Journal of Combinatorial Theory, Series B, 96 (6): 846–854, doi:10.1016/j.jctb.2006.02.010, MR 2274079

handle.net

hdl.handle.net

Halin (1964). Halin, Rudolf (1964), "Über unendliche Wege in Graphen", Mathematische Annalen, 157 (2): 125–137, doi:10.1007/bf01362670, hdl:10338.dmlcz/102294, MR 0170340, S2CID 122125458

More precisely, in the original formulation of this result by Halin (1964) in which ends are defined as equivalence classes of rays, every equivalence class of rays of

G

contains a unique nonempty equivalence class of rays of the spanning forest. In terms of havens, there is a one-to-one correspondence of havens of order

\aleph _{0}

between

G

and its spanning tree

T

for which

\beta _{T}(X)\subset \beta _{G}(X)

for every finite set

X

and every corresponding pair of havens

\beta _{T}

and

\beta _{G}

. Halin, Rudolf (1964), "Über unendliche Wege in Graphen", Mathematische Annalen, 157 (2): 125–137, doi:10.1007/bf01362670, hdl:10338.dmlcz/102294, MR 0170340, S2CID 122125458

semanticscholar.org

api.semanticscholar.org

Halin (1964). Halin, Rudolf (1964), "Über unendliche Wege in Graphen", Mathematische Annalen, 157 (2): 125–137, doi:10.1007/bf01362670, hdl:10338.dmlcz/102294, MR 0170340, S2CID 122125458

More precisely, in the original formulation of this result by Halin (1964) in which ends are defined as equivalence classes of rays, every equivalence class of rays of

G

contains a unique nonempty equivalence class of rays of the spanning forest. In terms of havens, there is a one-to-one correspondence of havens of order

\aleph _{0}

between

G

and its spanning tree

T

for which

\beta _{T}(X)\subset \beta _{G}(X)

for every finite set

X

and every corresponding pair of havens

\beta _{T}

and

\beta _{G}

. Halin, Rudolf (1964), "Über unendliche Wege in Graphen", Mathematische Annalen, 157 (2): 125–137, doi:10.1007/bf01362670, hdl:10338.dmlcz/102294, MR 0170340, S2CID 122125458

End (graph theory) (English Wikipedia)

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