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Characterization of general position sets and its applications to cographs and bipartite graphs

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  • Anand, Bijo S.
  • Chandran S. V., Ullas
  • Changat, Manoj
  • Klavžar, Sandi
  • Thomas, Elias John

Abstract

A vertex subset S of a graph G is a general position set of G if no vertex of S lies on a geodesic between two other vertices of S. The cardinality of a largest general position set of G is the general position number gp(G) of G. It is proved that S⊆V(G) is in general position if and only if the components of G[S] are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of S. If diam(G)=2, then gp(G) is the maximum of ω(G) and the maximum order of an induced complete multipartite subgraph of the complement of G. As a consequence, gp(G) of a cograph G can be determined in polynomial time. If G is bipartite, then gp(G) ≤ α(G) with equality if diam(G) ∈ {2, 3}. A formula for the general position number of the complement of an arbitrary bipartite graph is deduced and simplified for the complements of trees, of grids, and of hypercubes.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:359:y:2019:i:c:p:84-89
    DOI: 10.1016/j.amc.2019.04.064
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    References listed on IDEAS

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    1. Kexiang Xu & Haiqiong Liu & Kinkar Ch. Das & Sandi Klavžar, 2018. "Embeddings into almost self-centered graphs of given radius," Journal of Combinatorial Optimization, Springer, vol. 36(4), pages 1388-1410, November.
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    Cited by:

    1. Di Stefano, Gabriele, 2022. "Mutual visibility in graphs," Applied Mathematics and Computation, Elsevier, vol. 419(C).
    2. 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).
    3. 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).
    4. Klavžar, Sandi & Rus, Gregor, 2021. "The general position number of integer lattices," Applied Mathematics and Computation, Elsevier, vol. 390(C).

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