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The Simultaneous Representation Problem for Chordal, Comparability and Permutation Graphs

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Algorithms and Data Structures (WADS 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5664))

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Abstract

We introduce the simultaneous representation problem, defined for any graph class \(\cal C\) characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G 1 and G 2, sharing some vertices X (and the corresponding induced edges), are said to have a simultaneous representation with respect to a graph class \(\cal C\), if there exist representations R 1 and R 2 of G 1 and G 2 that are “consistent” on X. Equivalently (for the classes \(\cal C\) that we consider) there exist edges E′ between G 1 − X and G 2 − X such that G 1 ∪ G 2 ∪ E′ belongs to class \(\cal C\).

Simultaneous representation problems are related to graph sandwich problems, probe graph recognition problems and simultaneous planar embeddings and have applications in any situation where it is desirable to consistently represent two related graphs.

In this paper we give efficient algorithms for the simultaneous representation problem on chordal, comparability and permutation graphs. These results complement the recent poly-time algorithms for recognizing probe graphs for the above classes and imply that the graph sandwich problem for these classes is solvable for an interesting special case: when the set of optional edges induce a complete bipartite graph. Moreover for comparability and permutation graphs, our results can be extended to solve a generalized version of the simultaneous representation problem when there are k graphs any two of which share a common vertex set X. This generalized version is equivalent to the graph sandwich problem when the set of optional edges induce a k-partite graph.

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References

  1. Berry, A., Golumbic, M.C., Lipshteyn, M.: Two tricks to triangulate chordal probe graphs in polynomial time. In: SODA 2004: Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 992–969 (2004)

    Google Scholar 

  2. Berry, A., Golumbic, M.C., Lipshteyn, M.: Recognizing chordal probe graphs and cycle-bicolorable graphs. SIAM J. Discret. Math. 21(3), 573–591 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brass, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D.P., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Comput. Geom. Theory Appl. 36(2), 117–130 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chandler, D.B., Chang, M.-S., Kloks, T., Liu, J., Peng, S.-L.: On probe permutation graphs (extended abstract). In: Cai, J.-Y., Cooper, S.B., Li, A. (eds.) TAMC 2006. LNCS, vol. 3959, pp. 494–504. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  5. Chandler, D.B., Chang, M.-S., Kloks, T., Liu, J., Peng, S.-L.: Partitioned probe comparability graphs. Theor. Comput. Sci. 396(1-3), 212–222 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Estrella-Balderrama, A., Gassner, E., Jünger, M., Percan, M., Schaefer, M., Schulz, M.: Simultaneous geometric graph embeddings. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 280–290. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  7. Even, S., Pnueli, A., Lempel, A.: Permutation graphs and transitive graphs. J. Assoc. Comput. Mach. 19, 400–410 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. Golumbic, M.C.: Algorithmic Graph Theory And Perfect Graphs. Academic Press, New York (1980)

    MATH  Google Scholar 

  9. Golumbic, M.C., Kaplan, H., Shamir, R.: Graph sandwich problems. J. Algorithms 19(3), 449–473 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Golumbic, M.C., Lipshteyn, M.: Chordal probe graphs. Discrete Appl. Math. 143(1-3), 221–237 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jampani, K.R., Lubiw, A.: Simultaneous membership problem for chordal, comparability and permutation graphs (2009)

    Google Scholar 

  12. McConnell, R.M., Spinrad, J.P.: Construction of probe interval models. In: SODA 2002: Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, Philadelphia, PA, USA, pp. 866–875. Society for Industrial and Applied Mathematics (2002)

    Google Scholar 

  13. McMorris, F.R., Wang, C., Zhang, P.: On probe interval graphs. Discrete Appl. Math. 88(1-3), 315–324 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jampani, K.R.: Simultaneous membership and related problems, Ph.D thesis, University of Waterloo (in progress)

    Google Scholar 

  15. Spinrad, J.: Efficient Graph Representations. Fields Institute Monographs. American Mathematical Society (2003)

    Google Scholar 

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Author information

Authors and Affiliations

  1. David R. Cheriton School of Computer Science, University of Waterloo, Ontario, Canada

    Krishnam Raju Jampani & Anna Lubiw

Authors
  1. Krishnam Raju Jampani
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  2. Anna Lubiw
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Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. Department of Computer Science, University of Calgary, 2500 University Drive N.W., T2N 1N4, Calgary, AB, Canada

    Marina Gavrilova

  3. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, P.O. Box, K1S 5B6, Ontario, Canada

    Jörg-Rüdiger Sack

  4. University of Calgary, Calgary, AB, Canada

    Csaba D. Tóth

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© 2009 Springer-Verlag Berlin Heidelberg

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Jampani, K.R., Lubiw, A. (2009). The Simultaneous Representation Problem for Chordal, Comparability and Permutation Graphs. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_34

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  • DOI: https://doi.org/10.1007/978-3-642-03367-4_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03366-7

  • Online ISBN: 978-3-642-03367-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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