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Strong geodetic cores and Cartesian product graphs

Author

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  • Gledel, Valentin
  • Iršič, Vesna
  • Klavžar, Sandi

Abstract

The strong geodetic problem on a graph G is to determine a smallest set of vertices such that by fixing one shortest path between each pair of its vertices, all vertices of G are covered. To do this as efficiently as possible, strong geodetic cores and related numbers are introduced. Sharp upper and lower bounds on the strong geodetic core number are proved. Using the strong geodetic core number, an earlier upper bound on the strong geodetic number of Cartesian products is improved. It is also proved that sg(G □ K2) ≥ sg(G) holds for different families of graphs, a result conjectured to be true in general. Counterexamples are constructed demonstrating that the conjecture does not hold in general.

Suggested Citation

  • Gledel, Valentin & Iršič, Vesna & Klavžar, Sandi, 2019. "Strong geodetic cores and Cartesian product graphs," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    • Handle: RePEc:eee:apmaco:v:363:y:2019:i:c:45
    DOI: 10.1016/j.amc.2019.124609
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    References listed on IDEAS

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    1. Li, Zhao & Zuo, Liancui, 2018. "The k-path vertex cover in Cartesian product graphs and complete bipartite graphs," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 69-79.
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