Skip to main content
×
Home
    • Aa
    • Aa

Some natural decision problems in automatic graphs

  • Dietrich Kuske (a1) and Markus Lohrey (a2)
Abstract
Abstract

For automatic and recursive graphs, we investigate the following problems:

(A) existence of a Hamiltonian path and existence of an infinite path in a tree

(B) existence of an Euler path, bounding the number of ends, and bounding the number of infinite branches in a tree

(C) existence of an infinite clique and an infinite version of set cover

The complexity of these problems is determined for automatic graphs and. supplementing results from the literature, for recursive graphs. Our results show that these problems

(A) are equally complex for automatic and for recursive graphs (-complete).

(B) are moderately less complex for automatic than for recursive graphs (complete for different levels of the arithmetic hierarchy),

(C) are much simpler for automatic than for recursive graphs (decidable and -complete, resp.).

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[3]D. R. Bean , Recursive Euler and Hamilton paths, Proceedings of the American Mathematical Society, vol. 55 (1976), no. 2, pp. 385394.

[4]C. H. Bennett , Logical reversibility of computation, IBM Journal of Research and Development, vol. 17 (1973), pp. 525532.

[8]A. Blumensath and E. Grädel , Finite presentations of infinite structures: Automata and interpretations, Theory of Computing Systems, vol. 37 (2004), no. 6, pp. 641674.

[11]P. Erdös , T. Grünwald , and E. Vazsonyi , Über Euler–Linien unendlicher Graphen, Journal of Mathematics and Physics, vol. 17 (1938), no. 2, pp. 5975.

[13]M. R. Garey , D. S. Johnson , and R. E. Tarjan , The planar Hamiltonian circuit problem is NP-complete, SIAM Journal on Computing, vol. 5 (1976), no. 4, pp. 704714.

[17]D. Harel , Effective transformations on infinite trees, with applications to high undecidability, dominoes, and fairness, Journal of the Association for Computing Machinery, vol. 33 (1986), no. 1, pp. 224248.

[18]D. Harel , Hamiltonian paths in infinite graphs, Israel Journal of Mathematics, vol. 76 (1991), no. 3, pp. 317336.

[19]T. Hirst and D. Harel , Taking it to the limit: on infinite variants of NP-complete problems, Journal of Computer and System Sciences, vol. 53 (1996), pp. 180193.

[22]B. Khoussainov , A. Nies , S. Rubin , and F. Stephan , Automatic structures: richness and limitations. Logical Methods in Computer Science, vol. 3 (2007), no. 2, 2:2, 18 pp. (electronic).

[24]B. Khoussainov , S. Rubin , and F. Stephan , Definability and regularity in automatic structures, STACS 2004, Lecture Notes in Computer Science, no. 2996, Springer, 2004, pp. 440451.

[25]B. Khoussainov , S. Rubin , and F. Stephan , Automatic linear orders and trees, ACM Transactions on Computational Logic, vol. 6 (2005), no. 4, pp. 675700.

[26]S. C. Kleene , Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 4173.

[31]O. Ly , Automatic graphs and D0L-sequences of finite graphs, Journal of Computer and System Sciences, vol. 67 (2003), no. 3, pp. 497545.

[37]H. Wang , Proving theorems by pattern recognition, Bell Systems Technilogical Journal, vol. 40 (1961), pp. 141.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×