Abstract
We study left-hand side restrictions of the induced subgraph isomorphism problem: Fixing a class , for given graphs G and arbitrary H we ask for induced subgraphs of H isomorphic to G.
For the homomorphism problem this kind of restriction has been studied by Grohe and Dalmau, Kolaitis and Vardi for the decision problem and by Dalmau and Jonsson for its counting variant.
We give a dichotomy result for both variants of the induced subgraph isomorphism problem. Under some assumption from parameterized complexity theory, these problems are solvable in polynomial time if and only if contains no arbitrarily large graphs.
All classifications are given by means of parameterized complexity. The results are presented for arbitrary structures of bounded arity which implies, for example, analogous results for directed graphs.
Furthermore, we show that no such dichotomy is possible in the sense of classical complexity. That is, if there are classes such that the induced subgraph isomorphism problem on is neither in nor -complete. This argument may be of independent interest, because it is applicable to various parameterized problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)
Bodirski, M., Grohe, M.: Non-dichotomies in Constraint Satisfaction Complexity. In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2008, Track B) (to appear, 2008)
Chen, Y., Flum, J.: On parameterized path and chordless path problems. In: Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity, pp. 250–263. IEEE Computer Society, Los Alamitos (2007)
Dalmau, V., Jonsson, P.: The complexity of counting homomorphisms seen from the other side. Theor. Comput. Sci. 329(1-3), 315–323 (2004)
Dalmau, V., Kolaitis, P.G., Vardi, M.Y.: Constraint satisfaction, bounded treewidth, and finite-variable logics. In: Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming, CP 2002, pp. 310–326 (2002)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness I: Basic results. SIAM J. Comput. 24(4), 873–921 (1995)
Flum, J., Grohe, M.: Fixed-parameter tractability, definability, and model-checking. SIAM J. Comput. 31(1), 113–145 (2001)
Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33(4), 892–922 (2004)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1) (2007)
Ladner, R.E.: On the structure of polynomial time reducibility. J. ACM 22(1), 155–171 (1975)
McCartin, C.: Parameterized counting problems. Ann. Pure Appl. Logic 138(1-3), 147–182 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chen, Y., Thurley, M., Weyer, M. (2008). Understanding the Complexity of Induced Subgraph Isomorphisms. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_48
Download citation
DOI: https://doi.org/10.1007/978-3-540-70575-8_48
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70574-1
Online ISBN: 978-3-540-70575-8
eBook Packages: Computer ScienceComputer Science (R0)