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Some definitions of central structures

Part of the Lecture Notes in Mathematics book series (LNM,volume 1073)

Abstract

Standard measures of the centrality, or suitability as a site for a facility, of a vertex in a network include the eccentricity, branch weight, and distance sum. The nature of the facility to be constructed (such as a pipeline) could necessitate selecting a structure (such as a path) rather than just a point at which to locate the facility. Similarly, the facility may be required to "service" structures or areas within the network, and not just points. The same three measures of centrality can be applied to structures within the network, and it is proposed that four classes of locational problems should be considered : vertex-serves-vertex, vertex-serves-structure, structure-serves-vertex and, most generally, structure-serves-structure.

Keywords

  • Location Problem
  • Facility Location
  • Adjacent Vertex
  • Hamiltonian Path
  • Path Center

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. C. Berge, "Graphs and Hypergraphs," North Holland Publishing Co., Amsterdam, 1973.

    MATH  Google Scholar 

  2. E. J. Cockayne, S. T. Hedetniemi and S. L. Mitchell, Linear algorithms for finding the Jordan center and path center of a tree, Trans. Sci. 15, (1981), 98–114.

    CrossRef  MathSciNet  Google Scholar 

  3. P. M. Dearing, R. L. Francis and T. J. Lowe, Convex location problems on tree networks, Oper. Res., Vol. 24, No. 4, (1976), 628–642.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. A. J. Goldman, Optimal center location in simple networks, Trans. Sci. 5, (1971), 212–221.

    CrossRef  MathSciNet  Google Scholar 

  5. —, Minimax location of a facility in a network, Trans. Sci. 6, (1972), 407–418.

    CrossRef  MathSciNet  Google Scholar 

  6. S. L. Hakimi, Optimum locations of switching centers and the absolute centers and medians of a graph, Oper. Res. 12, (1964), 450–459.

    CrossRef  MATH  Google Scholar 

  7. S. L. Hakimi, E. F. Schmeichel and J. G. Pierce, On p-centers in networks, Trans. Sci. 12, (1978), 1–15.

    CrossRef  MathSciNet  Google Scholar 

  8. F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969.

    MATH  Google Scholar 

  9. C. Jordan, Sur les assemblages des lignes, J. Reine Angew. Math. 70, (1869), 185–190.

    CrossRef  MathSciNet  Google Scholar 

  10. E. Minieka, The m-center problem, SIAM Rev. 12, (1970), 138–139.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. —, The centers and medians of a graph, Oper. Res. 25, (1977), 641–650.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. C. A. Morgan and P. J. Slater, A linear algorithm for a core of a tree, J. Algorithms 1, (1980), 247–258.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. P. J. Slater, Centers to centroids in graphs, J. Graph Theory, Vol. 2, No. 3, (1978), 209–222.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. —, Structure of the k-centra of a tree, Proc. of the Ninth S.E. Conference on Combinatorics, Graph Theory and Computing, (1978), 663–670.

    Google Scholar 

  15. —, Locating Central paths in a graph, Trans. Sci. 16, (1982), 1–18.

    CrossRef  MathSciNet  Google Scholar 

  16. —, One-point location of an area response protection group, Sandia Laboratories Report, SAND 78-1788 (1978).

    Google Scholar 

  17. —, On locating a facility to service areas within a network, Oper. Res. 29, (1981), 523–531.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. B. Zelinka, Medians and peripherians of trees, Arch. Math., Brno (1968), 87–95.

    Google Scholar 

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© 1984 Springer-Verlag

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Slater, P.J. (1984). Some definitions of central structures. In: Koh, K.M., Yap, H.P. (eds) Graph Theory Singapore 1983. Lecture Notes in Mathematics, vol 1073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073115

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  • DOI: https://doi.org/10.1007/BFb0073115

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  • Print ISBN: 978-3-540-13368-1

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