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On Directed Tree Realizations of Degree Sets

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WALCOM: Algorithms and Computation (WALCOM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7748))

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Abstract

Given a degree set D = {a 1 < a 2 < … < a n } of non-negative integers, the minimum number of vertices in any tree realizing the set D is known [11]. In this paper, we study the number of vertices and multiplicity of distinct degrees as parameters of tree realizations of degree sets. We explore this in the context of both directed and undirected trees and asymmetric directed graphs. We show a tight lower bound on the maximum multiplicity needed for any tree realization of a degree set. For the directed trees, we study two natural notions of realizability by directed graphs and show tight lower bounds on the number of vertices needed to realize any degree set. For asymmetric graphs, if μ A (D) denotes the minimum number of vertices needed to realize any degree set, we show that a 1 + a n  + 1 ≤ μ A (D) ≤ a n − 1 + a n  + 1. We also derive sufficiency conditions on a i ’s under which the lower bound is achieved.

We study the following related algorithmic questions. (1) Given a degree set D and a non-negative integer r (as 1r), test whether the set D can be realized by a tree of exactly μ T (D) + r number of vertices. We show that the problem is fixed parameter tractable under two natural parameterizations of |D| and r. We also study the variant of the problem : (2) Given a tree T, and a non-negative integer r (in unary), test whether there exists another tree T′ such that T′ has exactly r more vertices than T and has the same degree set as T. We study the complexity of the problem in the case of directed trees as well.

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References

  1. Arikati, R., Srinavasa, Maheshwari, A.: Realizing degrees sequences in parallel. SIAM Journal of Discrete Mathematics 9, 317–338 (1996)

    Article  MATH  Google Scholar 

  2. Chartrand, G., Lesniak, L., Roberts, J.: Degree sets for digraphs. Periodica Mathematica Hungarica 7, 77–85 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Erdös, P., Gallai, T.: Graphs wiyh prescribed degrees of vertices. Mat. Lapok 11, 264–274 (1960)

    Google Scholar 

  4. Fellows, M.R., Gaspers, S., Rosamond, F.A.: Parameterizing by the Number of Numbers. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 123–134. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  5. Fernau, H.: Parameterized algorithms: A graph-theoretic approach. Technical report, Universität Tübingen, Tübingen, Germany (2005)

    Google Scholar 

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

    Google Scholar 

  7. Gupta, G., Joshi, P., Tripathi, A.: Graphic sequences of trees and a problem of frobenius. Czechoslovak Mathematical Journal 57, 49–52 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guy, R.K.: Unsolved Problems in Number Theory, Unsolved Problems in Intuitive Mathematics, 3rd edn., vol. I. Springer, New York (2004)

    Book  Google Scholar 

  9. Louis Hakimi, S.: On realizability of a set of integers as degrees of the vertices of a linear graph i. SIAM Journal of Discrete Mathematics 10, 496–506 (1962)

    Google Scholar 

  10. Havel, V.: Eine bemerkung über die existenz der endlichen graphen. Ĉasopis Pêst. Mat. 80, 477–480 (1955)

    MathSciNet  MATH  Google Scholar 

  11. Kapoor, S.F., Polimeni, A.D., Wall, C.E.: Degree sets for graphs. Fundamental Mathematics 95, 189–194 (1977)

    MathSciNet  MATH  Google Scholar 

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Kumar, P., Jayalal Sarma, M.N., Sawlani, S. (2013). On Directed Tree Realizations of Degree Sets. In: Ghosh, S.K., Tokuyama, T. (eds) WALCOM: Algorithms and Computation. WALCOM 2013. Lecture Notes in Computer Science, vol 7748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36065-7_26

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  • DOI: https://doi.org/10.1007/978-3-642-36065-7_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36064-0

  • Online ISBN: 978-3-642-36065-7

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