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On the mutual visibility in Cartesian products and triangle-free graphs

Author

Listed:
  • Cicerone, Serafino
  • Di Stefano, Gabriele
  • Klavžar, Sandi

Abstract

Given a graph G=(V(G),E(G)) and a set P⊆V(G), the following concepts have been recently introduced: (i) two elements of P are mutually visible if there is a shortest path between them without further elements of P; (ii)P is a mutual-visibility set if its elements are pairwise mutually visible; (iii) the mutual-visibility number of G is the cardinality of any largest mutual-visibility set. In this work we continue to investigate about these concepts. We first focus on mutual-visibility in Cartesian products. For this purpose, too, we introduce and investigate independent mutual-visibility sets. In the very special case of the Cartesian product of two complete graphs the problem is shown to be equivalent to the well-known Zarenkiewicz’s problem. We also characterize the triangle-free graphs with the mutual-visibility number equal to 3.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006920
    DOI: 10.1016/j.amc.2022.127619
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    References listed on IDEAS

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    1. Tian, Jing & Xu, Kexiang, 2021. "The general position number of Cartesian products involving a factor with small diameter," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    2. Klavžar, Sandi & Rus, Gregor, 2021. "The general position number of integer lattices," Applied Mathematics and Computation, Elsevier, vol. 390(C).
    3. Anand, Bijo S. & Chandran S. V., Ullas & Changat, Manoj & Klavžar, Sandi & Thomas, Elias John, 2019. "Characterization of general position sets and its applications to cographs and bipartite graphs," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 84-89.
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    Citations

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    Cited by:

    1. Ting Wang & Yu Jiang & Jianye Yang & Lei Xing, 2023. "Edge-Based Minimal k -Core Subgraph Search," Mathematics, MDPI, vol. 11(15), pages 1-17, August.
    2. Manuel, Paul & Brešar, Boštjan & Klavžar, Sandi, 2023. "Geodesic packing in graphs," Applied Mathematics and Computation, Elsevier, vol. 459(C).
    3. Brešar, Boštjan & Yero, Ismael G., 2024. "Lower (total) mutual-visibility number in graphs," Applied Mathematics and Computation, Elsevier, vol. 465(C).

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    1. Tian, Jing & Xu, Kexiang, 2021. "The general position number of Cartesian products involving a factor with small diameter," Applied Mathematics and Computation, Elsevier, vol. 403(C).
    2. Klavžar, Sandi & Rus, Gregor, 2021. "The general position number of integer lattices," Applied Mathematics and Computation, Elsevier, vol. 390(C).
    3. Di Stefano, Gabriele, 2022. "Mutual visibility in graphs," Applied Mathematics and Computation, Elsevier, vol. 419(C).

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