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Geodesic packing in graphs

Author

Listed:
  • Manuel, Paul
  • Brešar, Boštjan
  • Klavžar, Sandi

Abstract

A geodesic packing of a graph G is a set of vertex-disjoint maximal geodesics. The maximum cardinality of a geodesic packing is the geodesic packing number gpack(G). It is proved that the decision version of the geodesic packing number is NP-complete. We also consider the geodesic transversal number, gt(G), which is the minimum cardinality of a set of vertices that hit all maximal geodesics in G. While gt(G)≥gpack(G) in every graph G, the quotient gt(G)/gpack(G) is investigated. By using the rook's graph, it is proved that there does not exist a constant C<3 such that gt(G)gpack(G)≤C would hold for all graphs G. If T is a tree, then it is proved that gpack(T)=gt(T), and a linear algorithm for determining gpack(T) is derived. The geodesic packing number is also determined for the strong product of paths.

Suggested Citation

  • Manuel, Paul & Brešar, Boštjan & Klavžar, Sandi, 2023. "Geodesic packing in graphs," Applied Mathematics and Computation, Elsevier, vol. 459(C).
    • Handle: RePEc:eee:apmaco:v:459:y:2023:i:c:s0096300323004460
    DOI: 10.1016/j.amc.2023.128277
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    References listed on IDEAS

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    1. Di Stefano, Gabriele, 2022. "Mutual visibility in graphs," Applied Mathematics and Computation, Elsevier, vol. 419(C).
    2. Manuel, Paul & Brešar, Boštjan & Klavžar, Sandi, 2022. "The geodesic-transversal problem," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    3. Cicerone, Serafino & Di Stefano, Gabriele & Klavžar, Sandi, 2023. "On the mutual visibility in Cartesian products and triangle-free graphs," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    4. Iztok Peterin & Gabriel Semanišin, 2021. "On the Maximal Shortest Paths Cover Number," Mathematics, MDPI, vol. 9(14), pages 1-10, July.
    Full references (including those not matched with items on IDEAS)

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