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The general position number of Cartesian products involving a factor with small diameter

Author

Listed:
  • Tian, Jing
  • Xu, Kexiang

Abstract

A vertex subset R of a graph G is called a general position set if any triple V0⊆R is non-geodesic, this is, the three elements of V0 do not lie on the same geodesic in G. The general position number (gp-number for short) gp(G) of G is the number of vertices in a largest general position set in G. In this paper we first determine some formulae for the gp-numbers of Cartesian products involving a complete graph and of the Cartesian product of a complete multipartite graph with a path, respectively. Moreover, it is proved that gp(G□H)≤n(G)+n(H)−2 for any Cartesian product G□H with equality holding if and only if G and H are both generalized complete graphs, that is, a special class of graphs with diameters at most 2. Finally several open problems are proposed on the gp-numbers of Cartesian products.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:403:y:2021:i:c:s0096300321002964
    DOI: 10.1016/j.amc.2021.126206
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    References listed on IDEAS

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    1. 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.
    2. Klavžar, Sandi & Rus, Gregor, 2021. "The general position number of integer lattices," Applied Mathematics and Computation, Elsevier, vol. 390(C).
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    Cited by:

    1. 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).

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