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On the Maximal Shortest Paths Cover Number

Author

Listed:
  • Iztok Peterin

    (Institute of Mathematics and Physics, Faculty of Electrical Engineering and Computer Science, University of Maribor, 2000 Maribor, Slovenia)

  • Gabriel Semanišin

    (Institute of Computer Science, Faculty of Science, Pavol Jozef Šafárik University, 041 54 Košice, Slovakia)

Abstract

A shortest path P of a graph G is maximal if P is not contained as a subpath in any other shortest path. A set S ⊆ V ( G ) is a maximal shortest paths cover if every maximal shortest path of G contains a vertex of S . The minimum cardinality of a maximal shortest paths cover is called the maximal shortest paths cover number and is denoted by ξ ( G ) . We show that it is NP-hard to determine ξ ( G ) . We establish a connection between ξ ( G ) and several other graph parameters. We present a linear time algorithm that computes exact value for ξ ( T ) of a tree T .

Suggested Citation

  • Iztok Peterin & Gabriel Semanišin, 2021. "On the Maximal Shortest Paths Cover Number," Mathematics, MDPI, vol. 9(14), pages 1-10, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:14:p:1592-:d:589785
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    References listed on IDEAS

    as
    1. András Sebő & Eric Tannier, 2004. "On Metric Generators of Graphs," Mathematics of Operations Research, INFORMS, vol. 29(2), pages 383-393, May.
    2. Brešar, Boštjan & Kos, Tim & Krivoš-Belluš, Rastislav & Semanišin, Gabriel, 2019. "Hitting subgraphs in P4-tidy graphs," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 211-219.
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    Cited by:

    1. Manuel, Paul & Brešar, Boštjan & Klavžar, Sandi, 2022. "The geodesic-transversal problem," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    2. Manuel, Paul & Brešar, Boštjan & Klavžar, Sandi, 2023. "Geodesic packing in graphs," Applied Mathematics and Computation, Elsevier, vol. 459(C).

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