Abstract
This article concentrates on classes of graphs containing large grids and having a very regular structure. Grid-structured hierarchical graphs are defined in [19] by giving a static graph defining the content of a cell of a d-dimensional grid, repeating this static graph in each cell and by connecting the vertices in cells of a local neighborhood corresponding to a finite transit function in a uniform way. It is shown that for each finitely represented class K of dynamic graphs all monotone graph properties and all first order (FO) problems can be solved in constant time O(1). This result improves the linear time computability of FO problems for graphs of bounded degree from [25].
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Arnborg, S., Lagergren, J., Seese, D.: Easy Problems for Tree-Decomposable Graphs. Journal of Algorithms 12, 308–340 (1991)
Bodlaender, H.L., Koster, A.M.C.A.: Combinatorial Optimization on Graphs of bounded treewidth. The Computer Journal, special Volume on FPT, pp. 1–26, December 2006 (to appear)
Courcelle, B.: The monadic second order theory of graphs III: Tree decompositions, minors, and complexity issues. Theoret. Inform. Applic. 26(2), 257–286 (1992)
Diestel, R.: Graph Theory. Springer, Heidelberg (2000)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1997)
Ebbinghaus, H.-D., Flum, J., Thomas, W.: Mathematical Logic, 2nd edn. Springer, Heidelberg (1994)
Fagin, R., Stockmeyer, L., Vardi, M.: On monadic NP vs. monadic co-NP. In: The Proceedings of the 8th Annual IEEE Conference on Structure in Complexity Theory, pp. 19–30 (1993) (Full paper in Information and Computation 120(1), 78–92 (1995)
Frick, M., Grohe, M.: Deciding first-order properties of locally tree-decomposable structures. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, Springer, Heidelberg (1999)
Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. In: Proceedings of the 17th IEEE Symposium on Logic in Computer Science (LICS 2002), pp. 215–224 (2002)
Garey, M.R., Johnson, D.S.: Computers and Intractability. Bell Tel. Laboratories (1979)
Goldwasser, J., Klostermeyer, W., Trapp, G.: Characterizing Switch-Setting Problems. Linear and Multilinear Algebra 43, 121–135 (1997)
Hanf, W.: Model-theoretic methods in the study of elementary logic. In: Addison, J.W., Henkin, L.A., Tarski, A. (eds.) Symposium on the Theory of Models, pp. 132–145. North-Holland Publ. Co., Amsterdam (1965)
Hare, E.O., Hedetniemi, S.T., Hare, W.R.: Algorithms for Computing the domination number of KxN complete grid graphs. Congr. Numerantium 55, 81–92 (1986)
Hlineny, P., Oum, S., Seese, D., Gottlob, G.: Width Parameters Beyond Tree-width and Their Applications. The Computer Journal, spec. vol. on FPT, pp. 1–71(to appear)
Hoefting, F., Lengauer, T., Wanke, E.: Processing of hierarchically defined graphs and graph families. In: Monien, B., Ottmann, T. (eds.) Data Structures and Efficient Algorithms. LNCS, vol. 594, pp. 44–69. Springer, Heidelberg (1992)
Immerman, N.: Descriptive complexity. Springer, New York (1999)
Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamiltonian paths in grid graphs. SIAM Journal of Comp. 11(4), 676–686 (1982)
Iwano, K., Steiglitz, K.: Planarity testing of double periodic infinite graphs. Networks 18(3), 205–222 (1988)
Karp, R.M., Miller, R.E., Winograd, A.: The organization of computations for uniform recurrence equations. Journal of the ACM 14(3), 563–590 (1967)
Kaufmann, M., Mehlhorn, K.: Routing problems in grid graphs. In: Korte, B., Lovasz, L., Promel, H.J., Schrijver, A. (eds.) Paths, Flows and VLSI-Layout, pp. 165–184. Springer, Heidelberg (1990)
Lengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. B.G.Teubner and John Wiley & Sons, Chichester (1990)
Mitchell, W.F.: Hamiltonian paths through two- and three-dimensional grids. Journal of Research of the National Institute of Standards and Technology 110(2), 127–136 (2005)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)
Seese, D.: Tree-partite graphs and the complexity of algorithms (extended abstract). In: Budach, L. (ed.) FCT 1985. LNCS, vol. 199, pp. 412–421. Springer, Heidelberg (1985)
Seese, D.: Linear time computable problems and first-order descriptions. Math. Struct. in Comp. Science, 505–526 (1996)
Wagner, K.: The complexity of problems concerning graphs with regularities. In: Chytil, M.P., Koubek, V. (eds.) Mathematical Foundations of Computer Science 1984. LNCS, vol. 176, pp. 544–552. Springer, Heidelberg (1984) (extended abstract appeared in Proceedings of the 11th. MFCS)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Seese, D. (2007). Easy Problems for Grid-Structured Graphs. In: Preparata, F.P., Fang, Q. (eds) Frontiers in Algorithmics. FAW 2007. Lecture Notes in Computer Science, vol 4613. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73814-5_28
Download citation
DOI: https://doi.org/10.1007/978-3-540-73814-5_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73813-8
Online ISBN: 978-3-540-73814-5
eBook Packages: Computer ScienceComputer Science (R0)