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An Improved Nordhaus–Gaddum-Type Theorem for 2-Rainbow Independent Domination Number

Author

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  • Enqiang Zhu

    (Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China)

Abstract

For a graph G , its k -rainbow independent domination number, written as γ rik ( G ) , is defined as the cardinality of a minimum set consisting of k vertex-disjoint independent sets V 1 , V 2 , … , V k such that every vertex in V 0 = V ( G ) \ ( ∪ i = 1 k V i ) has a neighbor in V i for all i ∈ { 1 , 2 , … , k } . This domination invariant was proposed by Kraner Šumenjak, Rall and Tepeh (in Applied Mathematics and Computation 333(15), 2018: 353–361), which aims to compute the independent domination number of G □ K k (the generalized prism) via studying the problem of integer labeling on G . They proved a Nordhaus–Gaddum-type theorem: 5 ≤ γ ri 2 ( G ) + γ ri 2 ( G ¯ ) ≤ n + 3 for any n -order graph G with n ≥ 3 , in which G ¯ denotes the complement of G . This work improves their result and shows that if G ≇ C 5 , then 5 ≤ γ ri 2 ( G ) + γ ri 2 ( G ¯ ) ≤ n + 2 .

Suggested Citation

  • Enqiang Zhu, 2021. "An Improved Nordhaus–Gaddum-Type Theorem for 2-Rainbow Independent Domination Number," Mathematics, MDPI, vol. 9(4), pages 1-10, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:402-:d:501403
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    References listed on IDEAS

    as
    1. Kraner Šumenjak, Tadeja & Rall, Douglas F. & Tepeh, Aleksandra, 2018. "On k-rainbow independent domination in graphs," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 353-361.
    2. J. Amjadi & R. Khoeilar & M. Chellali & Z. Shao, 2020. "On the Roman domination subdivision number of a graph," Journal of Combinatorial Optimization, Springer, vol. 40(2), pages 501-511, August.
    3. J. Amjadi & R. Khoeilar & M. Chellali & Z. Shao, 0. "On the Roman domination subdivision number of a graph," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-11.
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