Graphs and Combinatorics

, Volume 34, Issue 3, pp 443–456 | Cite as

Strong Geodetic Number of Complete Bipartite Graphs and of Graphs with Specified Diameter

Original Paper
  • 46 Downloads

Abstract

The strong geodetic problem is a recent variation of the classical geodetic problem. For a graph G, its strong geodetic number \({{\mathrm{sg}}}(G)\) is the cardinality of a smallest vertex subset S, such that each vertex of G lies on one fixed shortest path between a pair of vertices from S. In this paper, some general properties of the strong geodetic problem are studied, especially in connection with the diameter of a graph. The problem is also solved for balanced complete bipartite graphs.

Keywords

Geodetic number Strong geodetic number Isometric path number Complete bipartite graphs Diameter 

Mathematics Subject Classification

05C12 05C70 

References

  1. 1.
    Ahangar, H.A., Kosari, S., Sheikholeslami, S.M., Volkmann, L.: Graphs with large geodetic number. Filomat 29, 1361–1368 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brešar, B., Kovše, M., Tepeh, A.: Geodetic sets in graphs. In: Structural Analysis of Complex Networks, pp. 197–218. Birkhäuser/Springer, New York (2011)Google Scholar
  3. 3.
    Centeno, C.C., Penso, L.D., Rautenbach, D., Pereira de Sá, V.G.: Geodetic number versus hull number in \(P_3\)-convexity. SIAM J. Discr. Math. 27, 717–731 (2013)CrossRefMATHGoogle Scholar
  4. 4.
    Chartrand, G., Harary, F., Zhang, P.: On the geodetic number of a graph. Networks 39, 1–6 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ekim, T., Erey, A.: Block decomposition approach to compute a minimum geodetic set. RAIRO Oper. Res. 48, 497–507 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ekim, T., Erey, A., Heggernes, P., van ’t Hof, P., Meister, D.: Computing minimum geodetic sets in proper interval graphs. Lect. Notes Comput. Sci. 7256, 279–290 (2012)CrossRefMATHGoogle Scholar
  7. 7.
    Fisher, D.C., Fitzpatrick, S.L.: The isometric path number of a graph. J. Combin. Math. Combin. Comput. 38, 97–110 (2001)MathSciNetMATHGoogle Scholar
  8. 8.
    Fitzpatrick, S.L.: Isometric path number of the Cartesian product of paths. Congr. Numer. 137, 109–119 (1999)MathSciNetMATHGoogle Scholar
  9. 9.
    Fraenkel, A.S., Harary, F.: Geodetic contraction games on graphs. Int. J. Game Theory 18, 327–338 (1989)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Harary, F., Loukakis, E., Tsouros, C.: The geodetic number of a graph. Math. Comput. Model. 17, 89–95 (1993)CrossRefMATHGoogle Scholar
  11. 11.
    Jiang, T., Pelayo, I.M., Pritikin, D.: Geodesic convexity and Cartesian products in graphs. Manuscript (2004)Google Scholar
  12. 12.
    Klavžar, S., Manuel, P.: Strong geodetic problem in grid like architectures. To appear Bull. Malays. Math. Sci. Soc.  https://doi.org/10.1007/s40840-018-0609-x
  13. 13.
    Lu, C.: The geodetic numbers of graphs and digraphs. Sci. Chin. Ser. A 50, 1163–1172 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Manuel, P., Klavžar, S., Xavier, A., Arokiaraj, A., Thomas, E.: Strong geodetic problem in networks: computational complexity and solution for Apollonian networks. Submitted (2016)Google Scholar
  15. 15.
    Manuel, P., Klavžar, S., Xavier, A., Arokiaraj, A., Thomas, E.: Strong edge geodetic problem in networks. Open Math. 15, 1225–1235 (2017)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pan, J.-J., Chang, G.J.: Isometric path numbers of graphs. Discr. Math. 306, 2091–2096 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pelayo, I.M.: Geodesic Convexity in Graphs. Springer Briefs in Mathematics. Springer, New York (2013)CrossRefMATHGoogle Scholar
  18. 18.
    Santhakumaran, A.P., John, J.: Edge geodetic number of a graph. J. Discr. Math. Sci. Cryprogr. 10, 415–432 (2007)MathSciNetMATHGoogle Scholar
  19. 19.
    Santhakumaran, A.P., John, J.: The connected edge geodetic number of a graph. Sci. Ser. A Math. Sci. (N.S.) 17, 67–82 (2009)MathSciNetMATHGoogle Scholar
  20. 20.
    Soloff, J.A., Márquez, R.A., Friedler, L.M.: Products of geodesic graphs and the geodetic number of products. Discuss. Math. Graph Theory 35, 35–42 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

Personalised recommendations