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Structural and Complexity Aspects of Line Systems of Graphs

  • Jozef Jirásek
  • Pavel Klavík
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6506)

Abstract

We study line systems in metric spaces induced by graphs. A line is a subset of vertices defined by a relation of betweenness.

We show that the class of all graphs having exactly k different lines is infinite if and only if it contains a graph with a bridge. We also study lines in random graphs—a random graph almost surely has \(n \choose 2\) different lines and no line containing all the vertices.

We call a pair of graphs isolinear if their line systems are isomorphic. We prove that deciding isolinearity of graphs is polynomially equivalent to the Graph Isomorphism Problem.

Similarly to the Graph Reconstruction Problem, we question the reconstructability of graphs from their line systems. We present a polynomial-time algorithm which constructs a tree from a given line system. We give an application of line systems: This algorithm can be extended to decide the existence of an embedding of a metric space into a tree metric and to construct this embedding if it exists.

Keywords

Bipartite Graph Random Graph Line System Chordal Graph Graph Isomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alon, N., Karp, R.M., Peleg, D., West, D.: A Graph-Theoretic Game and its Application to the k-Server Problem. SIAM J. Comput. 24(1), 78–100 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.: The Probabilistic Method. John Wiley, New York (1991)zbMATHGoogle Scholar
  3. 3.
    Bandelt, H.J.: Recognition of tree metrics. SIAM J. Discret. Math. 3(1), 1–6 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bandelt, H.J., Chepoi, V.: Metric graph theory and geometry: a survey. Contemporary Mathematics 453, 49–86 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bartal, Y.: On approximating arbitrary metrices by tree metrics. In: STOC 1998: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pp. 161–168. ACM, New York (1998)CrossRefGoogle Scholar
  6. 6.
    Bondy, J.A., Hemminger, R.L.: Graph reconstruction—a survey. Journal of Graph Theory 1(3), 227–268 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    de Bruijn, N.G., Erdős, P.: On a combinatorial problem. Indagationes Mathematicae 10, 421–423 (1948)zbMATHGoogle Scholar
  8. 8.
    Buneman, P.: A note on the metric properties of trees. Journal of Combinatorial Theory, Series B 17(1), 48–50 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, X., Chvátal, V.: Problems related to a de Bruijn-Erdős theorem. Discrete Appl. Math. 156(11), 2101–2108 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chiniforooshan, E., Chvátal, V.: A de Bruijn-Erdős theorem and metric spaces. arXiv:0906.0123v1 [math.CO] (2009)Google Scholar
  11. 11.
    Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Proc. Symp. Pure Math., vol. 7, pp. 101–179 (1963)Google Scholar
  12. 12.
    Golumbic, M.: Algorithmic Graph Theory and Perfect Graphs (2004)Google Scholar
  13. 13.
    Hemaspaandra, E., Hemaspaandra, L.A., Radziszowski, S.P., Tripathi, R.: Complexity results in graph reconstruction. Discrete Appl. Math. 155(2), 103–118 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kratsch, D., Hemaspaandra, L.A.: On the complexity of graph reconstruction. Math. Syst. Theory 27(3), 257–273 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lueker, G.S., Booth, K.S.: A linear time algorithm for deciding interval graph isomorphism. J. ACM 26(2), 183–195 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Menger, K.: Untersuchungen über allgemeine Metrik. Mathematische Annalen 100(1), 75–163 (1928)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zemlyachenko, V.N., Korneenko, N.M., Tyshkevich, R.I.: Graph isomorphism problem. Journal of Mathematical Sciences 29(4), 1426–1481 (1985)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jozef Jirásek
    • 1
  • Pavel Klavík
    • 1
  1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic

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