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Cleaning Interval Graphs

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Abstract

We investigate a special case of the Induced Subgraph Isomorphism problem, where both input graphs are interval graphs. We show the NP-hardness of this problem, and we prove fixed-parameter tractability of the problem with non-standard parameterization, where the parameter is the difference |V(G)|−|V(H)|, with G and H being the larger and the smaller input graph, respectively. Intuitively, we can interpret this problem as “cleaning” the graph G, regarded as a pattern containing extra vertices indicating errors, in order to obtain the graph H representing the original pattern. We also prove W[1]-hardness for the standard parameterization where the parameter is |V(H)|.

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Notes

  1. While we define modules in the standard way, our notion of closed modules is non-standard.

References

  1. Bodlaender, H.L.: Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees. J. Algorithms 11(4), 631–643 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13, 335–379 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai, L., Chan, S.M., Chan, S.O.: Random separation: A new method for solving fixed-cardinality optimization problems. In: IWPEC 2006: Proceedings of the 2nd International Workshop on Parameterized and Exact Computation. Lecture Notes in Computer Science, vol. 4169, pp. 239–250. Springer, Berlin (2006)

    Chapter  Google Scholar 

  4. Colbourn, C.J.: On testing isomorphism of permutation graphs. Networks 11, 13–21 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Colbourn, C.J., Booth, K.S.: Linear time automorphism algorithms for trees, interval graphs, and planar graphs. SIAM J. Comput. 10(1), 203–225 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Díaz, J., Thilikos, D.M.: Fast FPT-algorithms for cleaning grids. In: STACS 2006: Proceedings of the 23rd Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 3884, pp. 361–371. Springer, Berlin (2006)

    Google Scholar 

  7. Dinitz, Y., Itai, A., Rodeh, M.: On an algorithm of Zemlyachenko for subtree isomorphism. Inf. Process. Lett. 70(3), 141–146 (1999)

    Article  MathSciNet  Google Scholar 

  8. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)

    Book  Google Scholar 

  9. Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl. 3(3), 1–27 (1999)

    Article  MathSciNet  Google Scholar 

  10. Filotti, I.S., Mayer, J.N.: A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus (working paper). In: STOC 1980: Proceedings of the 12th Annual ACM Symposium on Theory of Computing, pp. 236–243. ACM, New York (1980)

    Chapter  Google Scholar 

  11. Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, New York (2006)

    Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences. Freeman, New York (1979)

    MATH  Google Scholar 

  13. Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Can. J. Math. 16, 539–548 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hajiaghayi, M., Nishimura, N.: Subgraph isomorphism, log-bounded fragmentation, and graphs of (locally) bounded treewidth. J. Comput. Syst. Sci. 73(5), 755–768 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lingas, A.: Subgraph isomorphism for biconnected outerplanar graphs in cubic time. Theor. Comput. Sci. 63(3), 295–302 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lueker, G.S., Booth, K.S.: A linear time algorithm for deciding interval graph isomorphism. J. ACM 26(2), 183–195 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Luks, E.M.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci. 25(1), 42–65 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Marx, D., Schlotter, I.: Parameterized graph cleaning problems. Discrete Appl. Math. 157(15), 3258–3267 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Matula, D.W.: Subtree isomorphism in O(n 5/2). Ann. Discrete Math. 2, 91–106 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Miller, G.L.: Isomorphism testing for graphs of bounded genus. In: STOC 1980: Proceedings of the 12th Annual ACM Symposium on Theory of Computing, pp. 225–235. ACM, New York (1980)

    Chapter  Google Scholar 

  21. Zemlyachenko, V.N.: Canonical numbering of trees. In: Proc. Seminar on Comb. Anal. at Moscow State University (1970). (In Russian)

    Google Scholar 

  22. Zemlyachenko, V.N.: Determining tree isomorphism. In: Voprosy Kibernetiki, Proc. of the Seminar on Combinatorial Mathematics, Moscow, 1971, pp. 54–60. Akad. Nauk SSSR, Scientific Council on the Complex Problem “Cybernetics”, 1973. (In Russian)

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Acknowledgement

Supported by the Hungarian National Research Fund OTKA 67651. Schlotter was also supported by the European Union and the European Social Fund (grant TÁMOP 4.2.1./B-09/1/KMR-2010-0003).

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Correspondence to Ildikó Schlotter.

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Marx, D., Schlotter, I. Cleaning Interval Graphs. Algorithmica 65, 275–316 (2013). https://doi.org/10.1007/s00453-011-9588-0

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