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Twilight graphs

Published online by Cambridge University Press:  12 March 2014

J. C. E. Dekker
J. C. E. Dekker*
Affiliation:
Institute for Advanced Study, Princeton, New Jersey 08540
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Abstract

This paper deals primarily with countable, simple, connected graphs and the following two conditions which are trivially satisfied if the graphs are finite:

(a) there is an edge-recognition algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to decide whether they are adjacent,

(b) there is a shortest path algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to find a minimal path joining them.

A graph G = ‹ν, η› with ν as set of vertices and η as set of edges is an α-graph if it satisfies (a); it is an ω-graph if it satisfies (b). G is called r.e. (isolic) if the sets ν and η are r.e. (isolated). While every ω-graph is an α-graph, the converse is false, even if G is r.e. or isolic. Several basic properties of finite graphs do not generalize to ω-graphs. For example, an ω-tree with more than one vertex need not have two pendant vertices, but may have only one or none, since it may be a 1-way or 2-way infinite path. Nevertheless, some elementary propositions for finite graphs G = ‹ν, η› with n = card(ν), e = card(η), do generalize to isolic ω-graphs, e.g., n − 1 ≤ e ≤ n(n − 1)/2. Let N and E be the recursive equivalence types of ν and η respectively. Then we have for an isolic α-tree G = ‹ν, η›: N = E + 1 iff G is an ω-tree. The theorem that every finite graph has a spanning tree has a natural, effective analogue for ω-graphs.

The structural difference between isolic α-graphs and isolic ω-graphs will be illustrated by: (i) every infinite graph is isomorphic to some isolic α-graph; (ii) there is an infinite graph which is not isomorphic to any isolic ω-graph.

An isolic α-graph is also called a twilight graph. There are c such graphs, c denoting the cardinality of the continuum.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 3 , September 1981 , pp. 539 - 571
Copyright
Copyright © Association for Symbolic Logic 1981

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References

REFERENCES

[1]Behzad, M. and Chartrand, G., Introduction to the theory of graphs, Allyn and Bacon, Boston, 1971.Google Scholar
[2]Dekker, J.C.E., Good choice sets, Annali della Scuola Normale Superiore de Pisa, Serie III, vol. 20(1966), pp. 367–393.Google Scholar
[3]Dekker, J.C.E., Regressive isols, Sets, models and recursion theory (Crossley, J.N., Editor), North-Holland, Amsterdam, 1967, pp. 272–296.CrossRefGoogle Scholar
[4]Dekker, J.C.E., Countable vector spaces with recurisive operations. Part I, this Journal, vol. 34 (1969), pp. 363–387.Google Scholar
[5]Dekker, J.C.E., Countable vector spaces with recursive operations. Part II, this Journal, vol. 36 (1971), pp. 477–493.Google Scholar
[6]Dekker, J.C.E., Projective bigraphs with recursive operations, Notre Dame Journal of Formal Logic, vol. 19 (1978), 193–199.CrossRefGoogle Scholar
[7]Dekker, J.C.E. and Myhill, J., Recursive equivalence types, University of California Publications in Mathematics (N.S.), vol. 3 (1960), pp. 67–214.Google Scholar
[8]Dekker, J.C.E. and Ellentuck, E., Recursion relative to regressive functions, Annals of Logic, vol. 6 (1974), pp. 231–257.CrossRefGoogle Scholar
[9]Harary, F., Graph theory, Addison-Wesley, Reading, Massachusetts, 1969.CrossRefGoogle Scholar
[10]König, D., Theorie der endlichen und unendlichen Graphen, Akademische Verlagsgesellschaft, Leipzig, 1936. Reprinted by Chelsea, New York, 1950.Google Scholar
[11]McLaughlin, T., Trees and isols. Part I, Rocky Mountain Journal of Mathematics, vol. 5 (1975), pp. 401–418.CrossRefGoogle Scholar
[12]McLaughlin, T., Trees and isols. Part II, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 22 (1976), pp. 45–78.CrossRefGoogle Scholar
[13]Nerode, A., Extensions to isols, Annals of Mathematics, vol. 73 (1961), pp. 362–403.CrossRefGoogle Scholar
[14]Nerode, A., Extensions to isolic integers, Annals of Mathematics, vol. 75 (1962), pp. 419–448.CrossRefGoogle Scholar