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Simplicial minors and decompositions of graphs

Published online by Cambridge University Press:  24 October 2008

Reinhard Diestel
Reinhard Diestel
Affiliation:
St. John's College, Cambridge
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Extract

The purpose of this paper is to give natural characterizations of the countable graphs that admit tree-decompositions or simplicial tree-decompositions into primes. Tree-decompositions were recently introduced by Robertson and Seymour in their series of papers on graph minors [7]. Simplicial tree-decompositions were first considered by Halin[6], being the most typical kind of ‘simplicial decomposition’ as introduced by Halin[5] in 1964. The problem of determining which infinite graphs admit a simplicial decomposition into primes has stood unresolved since then; a first solution for simplicial tree-decompositions was given in [2].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 3 , May 1988 , pp. 409 - 426
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Diestel, R.. Simplicial tree-decompositions of infinite graphs I. (Submitted.)Google Scholar
[2]Diestel, R.. Simplicial tree-decompositions of infinite graphs II—the existence of prime decompositions. (Submitted.)Google Scholar
[3]Diestel, R.. Simplicial tree-decompositions of infinite graphs III—the uniqueness of prime decompositions. (Submitted.)Google Scholar
[4]Dirac, G. A.. Simplicial decompositions of infinite graphs. In Combinatorial, Conv. Roma 1983, Symposia Math. vol. 28 (1986), pp. 159–195.Google Scholar
[5]Halin, R.. Simpliziale Zerfällungen beliebiger (endlicher oder unendlicher) Graphen. Math. Ann. 156 (1964), 216–225.CrossRefGoogle Scholar
[6]Halin, R.. On the representation of triangulated graphs in trees. Europ. J. Combinatorics 5 (1984), 23–28.CrossRefGoogle Scholar
[7]Robertson, N. and Seymour, P. D.. Graph minors I-XVIII. J. Combin. Theory B (in the Press).Google Scholar
[8]Wagner, K.. Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114 (1937), 570–590.CrossRefGoogle Scholar