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On the Complexity of the Balanced Vertex Ordering Problem

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Computing and Combinatorics (COCOON 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3595))

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Abstract

We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedlet al. [1]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but \(\mathcal{NP}\)-hard for graphs with maximum degree six. One of our main results is closing the gap in these results, by proving \(\mathcal{NP}\)-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains \(\mathcal{NP}\)-hard for planar graphs with maximum degree six and for 5-regular graphs. On the other hand we present a polynomial time algorithm that determines whether there is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an ‘almost balanced’ ordering.

Supported by grant MEC SB2003-0270. Research completed at the Department of Applied Mathematics and the Institute for Theoretical Computer Science, Charles University, Prague, Czech Republic. Supported by projects LN00A056 and 1M0021620808 of the Ministry of Education of the Czech Republic, and by the European Union Research Training Network COMBSTRU (Combinatorial Structure of Intractable Problems)

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References

  1. Biedl, T., Chan, T., Ganjali, Y., Hajiaghayi, M.T., Wood, D.R.: Balanced vertex-orderings of graphs. Discrete Applied Mathematics 148(1), 27–48 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16, 4–32 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kant, G., He, X.: Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theoretical Computer Science 172(1-2), 175–193 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Papakostas, A., Tollis, I.G.: Algorithms for area-efficient orthogonal drawings. Computational Geometry: Theory and Applications 9, 83–110 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of 10th Annual ACM Symposium on Theory of Computing (STOC 1978), pp. 216–226. ACM, New York (1978)

    Chapter  Google Scholar 

  6. Wood, D.R.: Minimizing the number of bends and volume in three-dimensional orthogonal graph drawings with a diagonal vertex layout. Algorithmica 39, 235–253 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wood, D.R.: Optimal three-dimensional orthogonal graph drawing in the general position model. Theoretical Computer Science 299(1-3), 151–178 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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© 2005 Springer-Verlag Berlin Heidelberg

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Kára, J., Kratochvíl, J., Wood, D.R. (2005). On the Complexity of the Balanced Vertex Ordering Problem. In: Wang, L. (eds) Computing and Combinatorics. COCOON 2005. Lecture Notes in Computer Science, vol 3595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11533719_86

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  • DOI: https://doi.org/10.1007/11533719_86

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28061-3

  • Online ISBN: 978-3-540-31806-4

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