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Planarity and minimal path algorithms

Part of: Computability and recursion theory

Published online by Cambridge University Press:  09 April 2009

Alfred B. Manaster ,
Jeffrey B. Remmel  and
James H. Schmerl
Alfred B. Manaster
Affiliation:
Department of Mathematics, University of Clifornia, San Diego, La Jolla, California 92093, U.S.A.
Jeffrey B. Remmel
Affiliation:
Department of Mathematics, University of Clifornia, San Diego, La Jolla, California 92093, U.S.A.
James H. Schmerl
Affiliation:
J.H. Schmerl, Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268, U.S.A.
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Abstract

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In 1981 two notions of effective presentation of countable connected graphs were formulated by J. C. E. Dekker—namely, edge recognition algorithm graphs and minimal path algorithm graphs. In this paper we show that every planar graph has a minimal path algorithm presentation but that some graphs have no minimal path algorithm presentations. We introduce the notion of a shortest distance algorithm graph, show that it lies strictly between the two notions of Dekker, and show that every graph has a shortest distance algorithm presentation. Finally, in contrast to Dekker's result about minimal path algorithm graphs, we produce a shortest distance algorithm graph which has no spanning tree which is an edge recognition algorithm graph.

Keywords

recursion theoryeffectively presented structuresgraph theory

MSC classification

Secondary: 03D45: Theory of numerations, effectively presented structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 1 , February 1986 , pp. 131 - 142
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Dekker, J. C. E., ‘Twilight graphs’, J. Symbolic Logic 46 (1981), 539–571.CrossRefGoogle Scholar
[2]Remmel, J. B., ‘Effective structures not contained in recursively enumerable sructures’, (pp. 206–226 in) Aspects of Effective Algebra, Crossley, J., ed. (Upside Down A Book Co., Clayton, Australia, 1979).Google Scholar