Towards an Isomorphism Dichotomy for Hereditary Graph Classes
In this paper we resolve the complexity of the isomorphism problem on
all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and algorithmic analysis of graphs. First, we develop a methodology to show isomorphism completeness of the isomorphism problem on graph classes by providing a general framework unifying various reduction techniques. Second, we generalize the concept of the modular decomposition to colored graphs, allowing for non-standard decompositions. We show that, given a suitable decomposition functor, the graph isomorphism problem reduces to checking isomorphism of colored prime graphs. Third, we extend the techniques of bounded color valence and hypergraph isomorphism on hypergraphs of bounded color class size as follows. We say a colored graph has generalized color valence at most k if, after removing all vertices in color classes of size at most k, for each color class C every vertex has at most k neighbors in C or at most k non-neighbors in C. We show that isomorphism of graphs of bounded generalized color valence can be solved in polynomial time.
graph isomorphism
modular decomposition
bounded color valence
reductions
forbidden induced subgraphs
689-702
Regular Paper
Pascal
Schweitzer
Pascal Schweitzer
10.4230/LIPIcs.STACS.2015.689
Vikraman Arvind, Bireswar Das, Johannes Köbler, and Seinosuke Toda. Colored hypergraph isomorphism is fixed parameter tractable. In FSTTCS, pages 327-337, 2010.
László Babai. Moderately exponential bound for graph isomorphism. In FCT, pages 34-50, 1981.
László Babai. Handbook of Combinatorics (vol. 2), chapter Automorphism groups, isomorphism, reconstruction, pages 1447-1540. MIT Press, 1995.
László Babai and Eugene M. Luks. Canonical labeling of graphs. In STOC, pages 171-183, 1983.
Kellogg S. Booth and C. J. Colbourn. Problems polynomially equivalent to graph isomorphism. Technical Report CS-77-04, Comp. Sci. Dep., Univ. Waterloo, 1979.
Andreas Brandstädt and Dieter Kratsch. On the structure of (P_5, gem)-free graphs. Discrete Applied Mathematics, 145(2):155-166, 2005.
Andrew Curtis, Min Lin, Ross McConnell, Yahav Nussbaum, Francisco Soulignac, Jeremy Spinrad, and Jayme Szwarcfiter. Isomorphism of graph classes related to the circular-ones property. Discrete Mathematics and Theoretical Computer Science, 15(1):157-182, 2013.
Konrad Dabrowski and Daniël Paulusma. Clique-width of graph classes defined by two forbidden induced subgraphs. CoRR, abs/1405.7092, 2014.
Konrad K. Dabrowski, Petr A. Golovach, and Daniel Paulusma. Colouring of graphs with ramsey-type forbidden subgraphs. Theoretical Computer Science, 522(0):34-43, 2014.
Frank Fuhlbrück. Fixed-parameter tractability of the graph isomorphism and canonization problems. Diploma thesis, Humboldt-Universität zu Berlin, 2013.
Mark K. Goldberg. A nonfactorial algorithm for testing isomorphism of two graphs. Discrete Applied Mathematics, 6(3):229-236, 1983.
Petr A. Golovach and Daniël Paulusma. List coloring in the absence of two subgraphs. Discrete Applied Mathematics, 166:123-130, 2014.
Martin Grohe and Dániel Marx. Structure theorem and isomorphism test for graphs with excluded topological subgraphs. In STOC, pages 173-192, 2012.
Yuri Gurevich. From invariants to canonization. Bulletin of the EATCS, 63, 1997.
Yuri Gurevich. From invariants to canonization. In Current Trends in Theoretical Computer Science, pages 327-331. World Scientific, 2001.
Michel Habib and Christophe Paul. A survey of the algorithmic aspects of modular decomposition. Computer Science Review, 4(1):41-59, 2010.
Tommi A. Junttila and Petteri Kaski. Conflict propagation and component recursion for canonical labeling. In TAPAS, pages 151-162, 2011.
Johannes Köbler, Uwe Schöning, and Jacobo Torán. The graph isomorphism problem: its structural complexity. Birkhäuser Verlag, Basel, Switzerland, 1993.
Johannes Köbler and Oleg Verbitsky. From invariants to canonization in parallel. In CSR, pages 216-227, 2008.
Daniel Král, Jan Kratochvíl, Zsolt Tuza, and Gerhard J. Woeginger. Complexity of coloring graphs without forbidden induced subgraphs. In WG, pages 254-262, 2001.
Stefan Kratsch and Pascal Schweitzer. Graph isomorphism for graph classes characterized by two forbidden induced subgraphs. In WG, pages 34-45, 2012.
Stefan Kratsch and Pascal Schweitzer. Graph isomorphism for graph classes characterized by two forbidden induced subgraphs. CoRR, abs/1208.0142, 2012.
Vadim V. Lozin. A decidability result for the dominating set problem. Theoretical Computer Science, 411(44-46):4023-4027, 2010.
Eugene M. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and System Sciences, 25(1):42-65, 1982.
Gary L. Miller. Isomorphism testing and canonical forms for k-contractable graphs (a generalization of bounded valence and bounded genus). In FCT, pages 310-327, 1983.
Yota Otachi and Pascal Schweitzer. Isomorphism on subgraph-closed graph classes: A complexity dichotomy and intermediate graph classes. In ISAAC, pages 111-118, 2013.
Michaël Rao. Decomposition of (gem,co-gem)-free graphs. Unpublished, available at http://www.labri.fr/perso/rao/publi/decompgemcogem.ps, 2007.
Pascal Schweitzer. Problems of unknown complexity: Graph isomorphism and Ramsey theoretic numbers. PhD thesis, Universität des Saarlandes, Germany, 2009.
Pascal Schweitzer. Towards an isomorphism dichotomy for hereditary graph classes. CoRR, abs/1411.1977, 2014.
Ákos Seress. Permutation Group Algorithms. Cambridge Tracts in Mathematics. Cambridge University Press, 2003.
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