ISAAC 2010: Algorithms and Computation pp 157-168

# Structural and Complexity Aspects of Line Systems of Graphs

• Jozef Jirásek
• Pavel Klavík
Conference paper
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.

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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)
2. 2.
Alon, N., Spencer, J.: The Probabilistic Method. John Wiley, New York (1991)
3. 3.
Bandelt, H.J.: Recognition of tree metrics. SIAM J. Discret. Math. 3(1), 1–6 (1990)
4. 4.
Bandelt, H.J., Chepoi, V.: Metric graph theory and geometry: a survey. Contemporary Mathematics 453, 49–86 (2008)
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)
6. 6.
Bondy, J.A., Hemminger, R.L.: Graph reconstruction—a survey. Journal of Graph Theory 1(3), 227–268 (1977)
7. 7.
de Bruijn, N.G., Erdős, P.: On a combinatorial problem. Indagationes Mathematicae 10, 421–423 (1948)
8. 8.
Buneman, P.: A note on the metric properties of trees. Journal of Combinatorial Theory, Series B 17(1), 48–50 (1974)
9. 9.
Chen, X., Chvátal, V.: Problems related to a de Bruijn-Erdős theorem. Discrete Appl. Math. 156(11), 2101–2108 (2008)
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)
14. 14.
Kratsch, D., Hemaspaandra, L.A.: On the complexity of graph reconstruction. Math. Syst. Theory 27(3), 257–273 (1994)
15. 15.
Lueker, G.S., Booth, K.S.: A linear time algorithm for deciding interval graph isomorphism. J. ACM 26(2), 183–195 (1979)
16. 16.
Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)
17. 17.
Menger, K.: Untersuchungen über allgemeine Metrik. Mathematische Annalen 100(1), 75–163 (1928)
18. 18.
Zemlyachenko, V.N., Korneenko, N.M., Tyshkevich, R.I.: Graph isomorphism problem. Journal of Mathematical Sciences 29(4), 1426–1481 (1985)