Abstract
The fixed-parameter tractability of graph isomorphism is an open problem with respect to a number of natural parameters, such as tree-width, genus and maximum degree. We show that graph isomorphism is fixed-parameter tractable when parameterized by the tree-depth of the graph. We also extend this result to a parameter generalizing both tree-depth and max-leaf-number by deploying new variants of cops-and-robbers games.
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References
Arvind, V., Das, B., Johannes, K., Toda, S.: Colored Hypergraph Isomorphism is Fixed Parameter Tractable. In: ECCC 93 (2009)
Babai, L.: Monte-Carlo algorithms in graph isomorphism testing. Tech. Rep. DMS 79-10, Université de Montréal, pp. 1–33 (1979)
Bodlaender, H., Hafsteinsson, H., Gilbert, J.R., Kloks, T.: Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms 18, 238–255 (1995)
Bodlaender, H.L.: Polynomial Algorithms for Graph lsomorphism Chromatic Index on Partial k-Trees. Journal of Algorithms 11(4), 631–643 (1990)
Cai, J.-Y., Fürer, M., Immerman, N.: An Optimal Lower Bound on the Number of Variables for Graph Identification. Combinatorica 12(4), 389–410 (1992)
Dvořák, Z., Giannopoulou, A., Thilikos, D.M.: Forbidden graphs for tree-depth. European Journal of Combinatorics 33(5), 969–979 (2012)
Elberfeld, M., Grohe, M.: Where First-Order and Monadic Second-Order Logic Coincide. Arxiv preprint arXiv:1204.6291, pp. 1–15 (2012)
Evdokimov, S., Ponomarenko, I.: Isomorphism of Coloured Graphs with Slowly Increasing Multiplicity of Jordan Blocks. Combinatorica 19(3), 321–333 (1999)
Fellows, M., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F., Saurabh, S.: The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number. Theory of Computing Systems 45(4), 822–848 (2009)
Furst, M., Hopcroft, J.: Luks: Polynomial-time algorithms for permutation groups. In: Proc. FOCS 1980, pp. 36–41 (1980)
Ganian, R., Hliněný, P., Kneis, J., Langer, A., Obdržálek, J., Rossmanith, P.: On Digraph Width Measures in Parameterized Algorithmics. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 185–197. Springer, Heidelberg (2009)
Giannopoulou, A., Hunter, P., Thilikos, D.: LIFO-search: A min-max theorem and a searching game for cycle-rank and tree-depth. Submitted to J. Discrete Math. (2011)
Grohe, M., Marx, D.: Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs. In: Proc. STOC 2012, pp. 173–192 (2012)
Grohe, M.: Fixed-point definability and polynomial time on graphs with excluded minors. In: Proc. LICS 2010, pp. 179–188 (2010)
Heath, M., Ng, E., Peyton, B.: Parallel algorithms for sparse linear systems. SIAM Review 33(3), 420–460 (1991)
Kleitman, D., West, D.: Spanning Trees with Many Leaves. SIAM J. Discrete Math. 4, 99–106 (1991)
Kratsch, S., Schweitzer, P.: Isomorphism for Graphs of Bounded Feedback Vertex Set Number. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 81–92. Springer, Heidelberg (2010)
Lindell, S.: A logspace algorithm for tree canonization. In: Proc. STOC 1992, pp. 400–404 (1992)
Lovász, L.: Graph minor theory. Bulletin of the AMS 43(1), 75–86 (2006)
Luks, E.: Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and System Sciences (1982)
Manne, F.: An Algorithm for Computing an Elimination Tree of Minimum Height for a Tree. Tech. Rep. CS-91-59, University of Bergen, Norway (1992)
Miller, G.: Isomorphism testing for graphs of bounded genus. In: Proc. STOC 1980, pp. 225–235 (1980)
Nešetřil, J., Ossona de Mendez, P.: Sparsity: Graphs, Structures and Algorithms. Algorithms and Combinatorics, vol. 28. Springer (2012)
Nešetřil, J., Ossona de Mendez, P.: Tree-depth, subgraph coloring and homomorphism bounds. European Journal of Combinatorics 27(6), 1022–1041 (2006)
Ponomarenko, I.: The isomorphism problem for classes of graphs that are invariant with respect to contraction. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 174, 147–177 (1988) (Russian)
Robertson, N., Seymour, P.: Graph minors XX. Wagners conjecture. Journal of Combinatorial Theory, Series B 92, 325–357 (2004)
Toda, S.: Computing automorphism groups of chordal graphs whose simplicial components are of small size. IEICE Transactions on Information and Systems E89-D(8), 2388–2401 (2006)
Yamazaki, K., Bodlaender, H.L., de Fluiter, B., Thilikos, D.M.: Isomorphism for Graphs of Bounded Distance Width. Algorithmica 24(2), 105–127 (1999)
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Bouland, A., Dawar, A., Kopczyński, E. (2012). On Tractable Parameterizations of Graph Isomorphism. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_21
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DOI: https://doi.org/10.1007/978-3-642-33293-7_21
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