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Steiner Convex Sets and Cartesian Product

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Abstract

In this paper we prove some bounds for Steiner distance in Cartesian product. We investigate properties of connected subgraphs that are not Steiner convex. Those results are the key in the characterization of Steiner convex sets of grids and also in the characterization of 3-Steiner convex sets of Cartesian product graphs.

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References

  1. van de Vel, M.L.J.: Theory of Convex Structures. North Holland, Amsterdam (1993)

    MATH  Google Scholar 

  2. Changat, M., Mulder, H.M., Sierksma, G.: Convexities related to path properties on graphs. Discret. Math. 290, 117–131 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bandelt, H.-J., Chepoi, V.: Metric Graph Theory and Geometry: A Survey, Surveys on Discrete and Computational Geometry, Contemp. Math., vol. 453, pp. 49–86. Amer. Math. Soc., Providence, RI (2008)

  4. Mulder, H.M.: The interval function of a graph. Mathematical Centre Tracts, vol. 132 Mathematisch Centrum, Amsterdam (1980)

  5. Brešar, B., Gologranc, T.: On a local 3-Steiner convexity. Eur. J. Comb. 32, 1222–1235 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cáceres, J., Oellermann, O.R.: On 3-Steiner simplicial orderings. Discret. Math. 309, 5828–5833 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Henning, M., Nielsen, M.H., Oellermann, O.R.: Local Steiner convexity. Eur. J. Comb. 30, 1186–1193 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Nielsen, M.H., Oellermann, O.R.: Steiner trees and convex geometries. SIAM J. Discret. Math. 23, 680–693 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gilbert, E.N., Pollak, H.O.: Steiner minimal trees. SIAM J. Appl. Math. 16, 1–29 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brešar, B., Changat, M., Mathews, J., Peterin, I., Narasimha-Shenoi, P.G., Tepeh Horvat, A.: Steiner intervals, geodesic intervals, and betweenness. Discret. Math. 309, 6114–6125 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cáceres, J., Marquez, A., Puertas, M.L.: Steiner distance and convexity in graphs. Eur. J. Comb. 29, 726–736 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kubicka, E., Kubicki, G., Oellermann, O.R.: Steiner intervals in graphs. Discret. Appl. Math. 81, 181–190 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Oellermann, O.R., Puertas, M.L.: Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs. Discret. Math. 307, 88–96 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gologranc, T.: Graphs with 4-Steiner convex balls. Taiwan. J. Math. 19, 1325–1340 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Anand, B.S., Changat, M., Klavžar, S., Peterin, I.: Convex sets in lexicographic products of graphs. Graphs Comb. 28, 77–84 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Canoy Jr., S.R., Garces, I.J.L.: Convex sets under some graph operations. Graphs Comb. 18, 787–793 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jiang, T., Pelayo, I., Pritikin, D.: Geodesic convexity and Cartesian products in graphs. Manuscript

  18. Peterin, I.: Intervals and convex sets in strong product of graphs. Graphs Comb. 29, 705–714 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Alcón, L., Brešar, B., Gologranc, T., Gutierrez, M., Kraner Šumenjak, T., Peterin, I., Tepeh, A.: Toll convexity. Eur. J. Comb. 46, 161–175 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs, 2nd edn. CRC Press, Boca Raton (2011)

    MATH  Google Scholar 

  21. Pelayo, I.M.: Geodesic Convexity in Graphs. Springer, New York (2013)

    Book  MATH  Google Scholar 

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Acknowledgments

This research was supported by the Slovenian Research Agency project L7–5459.

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Correspondence to Tanja Gologranc.

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Communicated by Sandi Klavžar.

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Gologranc, T. Steiner Convex Sets and Cartesian Product. Bull. Malays. Math. Sci. Soc. 41, 627–636 (2018). https://doi.org/10.1007/s40840-016-0312-8

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  • DOI: https://doi.org/10.1007/s40840-016-0312-8

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