Abstract
Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized by logarithmic-space (logspace) algorithms. This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous class of graphs of bounded tree width one of the few classes of graphs for which the complexity of the isomorphism problem has been exactly determined.
- Vikraman Arvind, Bireswar Das, and Johannes Köbler. 2008. A logspace algorithm for partial 2-tree canonization. In Proceedings of the 3rd International Computer Science Symposium in Russia (CSR’08) (Lecture Notes in Computer Science). Springer, 40--51. Google Scholar
Digital Library
- Vikraman Arvind, Bireswar Das, Johannes Köbler, and Sebastian Kuhnert. 2012. The isomorphism problem for -trees is complete for logspace. Info. Comput. 217 (2012), 1--11. Google Scholar
Digital Library
- László Babai. 2016. Graph isomorphism in quasipolynomial time {extended abstract}. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC’16). ACM, 684--697. Google Scholar
Digital Library
- Hans L. Bodlaender. 1990. Polynomial algorithms for graph isomorphism and chromatic index on partial -trees. J. Algor. 11, 4 (1990), 631--643. Google Scholar
Digital Library
- R. B. Boppana, J. Hastad, and S. Zachos. 1987. Does have short interactive proofs? Inf. Process. Lett. 25, 2 (May 1987), 127--132. Google Scholar
Digital Library
- Bireswar Das, MuraliKrishna Enduri, and I. Vinod Reddy. 2015. Logspace and algorithms for graph isomorphism for subclasses of bounded tree-width graphs. In Proceedings of the 9th International Workshop on Algorithms and Computation (WALCOM’15) (Lecture Notes in Computer Science), Vol. 8973. Springer, 329--334.Google Scholar
- Bireswar Das, Jacobo Torán, and Fabian Wagner. 2012. Restricted space algorithms for isomorphism on bounded treewidth graphs. Info. Comput. 217 (2012), 71--83. Google Scholar
Digital Library
- Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf, and Fabian Wagner. 2009a. Planar graph isomorphism is in log-space. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC’09). IEEE Computer Society, 203--214. Google Scholar
Digital Library
- Samir Datta, Prajakta Nimbhorka, Thomas Thierauf, and Fabian Wagner. 2009b. Graph isomorphism for -free and -free graphs is in log-space. In Proceedings of the 29th Annual IARCS Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’09) (Leibniz International Proceedings in Informatics), Vol. 4. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 145--156.Google Scholar
- Reinhard Diestel. 2005. Graph Theory (3rd ed.). Graduate Texts in Mathematics, Vol. 173. Springer.Google Scholar
- Michael Elberfeld, Andreas Jakoby, and Till Tantau. 2010. Logspace versions of the theorems of Bodlaender and Courcelle. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS’10). IEEE Computer Society, 143--152. Google Scholar
Digital Library
- Michael Elberfeld and Ken-ichi Kawarabayashi. 2014. Embedding and canonizing graphs of bounded genus in logspace. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC’14). ACM, New York, NY, 383--392. Google Scholar
Digital Library
- Michael Elberfeld and Pascal Schweitzer. 2016. Canonizing graphs of bounded tree width in logspace. In Proceedings of the 33rd Symposium on Theoretical Aspects of Computer Science (STACS’16) (Leibniz International Proceedings in Informatics), Vol. 47. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 32:1--32:14.Google Scholar
- Ion S. Filotti and Jack N. Mayer. 1980. A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus. In Proceedings of the 12th Annual ACM Symposium on Theory of Computing (STOC’80). ACM, New York, NY, 236--243. Google Scholar
Digital Library
- Martin Grohe. 2000. Isomorphism testing for embeddable graphs through definability. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC’00). ACM, 63--72. Google Scholar
Digital Library
- Martin Grohe. 2012. Fixed-point definability and polynomial time on graphs with excluded minors. J. ACM 59, 5 (2012), 27. Google Scholar
Digital Library
- Martin Grohe, Ken-ichi Kawarabayashi, and Bruce A. Reed. 2013. A simple algorithm for the graph minor decomposition—Logic meets structural graph theory. In Proceedings of the 23rd Annual ACM/SIAM Symposium on Discrete Algorithms (SODA’12). SIAM, 414--431. Google Scholar
Digital Library
- Martin Grohe and Oleg Verbitsky. 2006. Testing graph isomorphism in parallel by playing a game. In Proceedings of 33rd International Colloquium on Automata, Languages and Programming (ICALP’06) (Lecture Notes in Computer Science), Vol. 4051. Springer, 3--14. Google Scholar
Digital Library
- J. E. Hopcroft and J. K. Wong. 1974. Linear time algorithm for isomorphism of planar graphs (preliminary report). In Proceedings of the 6th Annual ACM Symposium on Theory of Computing (STOC’74). ACM, New York, NY, 172--184. Google Scholar
Digital Library
- Birgit Jenner, Johannes Köbler, Pierre McKenzie, and Jacobo Torán. 2003. Completeness results for graph isomorphism. J. Comput. Syst. Sci. 66, 3 (2003), 549--566. Google Scholar
Digital Library
- Neil D. Jones. 1975. Space-bounded reducibility among combinatorial problems. J. Comput. Syst. Sci. 11, 1 (1975), 68--85. Google Scholar
Digital Library
- Richard E. Ladner and Nancy A. Lynch. 1976. Relativization of questions about log space computability. Theory Comput. Syst. 10 (1976), 19--32. Issue 1.Google Scholar
- Hanns-Georg Leimer. 1993. Optimal decomposition by clique separators. Discrete Math. 113, 13 (1993), 99--123. Google Scholar
Digital Library
- Steven Lindell. 1992. A logspace algorithm for tree canonization (extended abstract). In Proceedings of the 24th Annual ACM Symposium on Theory of Computing (STOC’92). ACM, New York, NY, 400--404. Google Scholar
Digital Library
- Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. 2014a. Fixed-parameter tractable canonization and isomorphism test for graphs of bounded tree width. In Proceedings of the 55th IEEE Symposium on Foundations of Computer Science (FOCS’14). IEEE Computer Society, 186--195. Google Scholar
Digital Library
- Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. 2014b. Fixed-parameter tractable canonization and isomorphism test for graphs of bounded tree width. CoRR abs/1404.0818 (2014).Google Scholar
- Gary L. Miller. 1980. Isomorphism testing for graphs of bounded genus. In Proceedings of the 12th Annual ACM Symposium on Theory of Computing (STOC’80). ACM, New York, NY, 225--235. Google Scholar
Digital Library
- Yota Otachi and Pascal Schweitzer. 2014. Reduction techniques for graph isomorphism in the context of width parameters. In Proceedings of the 14th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT’14) (Lecture Notes in Computer Science), Vol. 8503. Springer, 368--379.Google Scholar
Cross Ref
- I. N. Ponomarenko. 1991. The isomorphism problem for classes of graphs closed under contraction. J. Math. Sci. 55, 2 (1991), 1621--1643.Google Scholar
Cross Ref
- Omer Reingold. 2008. Undirected connectivity in log-space. J. ACM 55, 4 (2008), 1--24. Google Scholar
Digital Library
- Uwe Schöning. 1988. Graph isomorphism is in the low hierarchy. J. Comput. Syst. Sci. 37, 3 (1988), 312--323. Google Scholar
Digital Library
- L. J. Stockmeyer and A. R. Meyer. 1973. Word problems requiring exponential time (preliminary report). In Proceedings of the 5th Annual ACM Symposium on Theory of Computing (STOC’73). ACM, 1--9. Google Scholar
Digital Library
- Robert Endre Tarjan. 1971. A V2 algorithm for determining isomorphism of planar graphs. Inform. Process. Lett. 1, 1 (1971), 32--34. Google Scholar
Digital Library
- Jacobo Torán. 2004. On the hardness of graph isomorphism. SIAM J. Comput. 33, 5 (2004), 1093--1108. Google Scholar
Digital Library
- Heribert Vollmer. 1999. Introduction to Circuit Complexity: A Uniform Approach. Springer, Berlin. Google Scholar
Digital Library
- Fabian Wagner. 2011. Graphs of bounded tree width can be canonized in AC1. In Proceedings of the 6th International Computer Science Symposium in Russia (CSR’11) (Lecture Notes in Computer Science). Springer, 209--222. Google Scholar
Digital Library
Index Terms
Canonizing Graphs of Bounded Tree Width in Logspace
Recommendations
An Improved Isomorphism Test for Bounded-tree-width Graphs
We give a new FPT algorithm testing isomorphism of n-vertex graphs of tree-width k in time 2kpolylog(k)n3, improving the FPT algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time 2O(k5 log k)n5. Based on an ...
Embedding and canonizing graphs of bounded genus in logspace
STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computingGraph embeddings of bounded Euler genus (that means, embeddings with bounded orientable or nonorientable genus) help to design time-efficient algorithms for many graph problems. Since linear-time algorithms are known to compute embeddings of any bounded ...
The isomorphism problem for k-trees is complete for logspace
We show that, for k constant, k-tree isomorphism can be decided in logarithmic space by giving an O(klogn) space canonical labeling algorithm. The algorithm computes a unique tree decomposition, uses colors to fully encode the structure of the original ...






Comments