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Complexity of k-rainbow independent domination and some results on the lexicographic product of graphs

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  • Brezovnik, Simon
  • Šumenjak, Tadeja Kraner

Abstract

A function f:V(G)→{0,1,…,k} is called a k-rainbow independent dominating function of G if Vi={x∈V(G):f(x)=i} is independent for 1 ≤ i ≤ k, and for every x ∈ V0 it follows that N(x) ∩ Vi ≠ ∅, for every i ∈ [k]. The k-rainbow independent domination number, γrik(G), of a graph G is the minimum of w(f)=∑i=1k|Vi| over all such functions. In this paper we show that the problem of determining whether a graph has a k-rainbow independent dominating function of a given weight is NP-complete for bipartite graphs and that there exists a linear-time algorithm to compute γrik(G) of trees. In addition, sharp bounds for the k-rainbow independent domination number of the lexicographic product are presented, as well as the exact formula in the case k=2.

Suggested Citation

  • Brezovnik, Simon & Šumenjak, Tadeja Kraner, 2019. "Complexity of k-rainbow independent domination and some results on the lexicographic product of graphs," Applied Mathematics and Computation, Elsevier, vol. 349(C), pages 214-220.
    • Handle: RePEc:eee:apmaco:v:349:y:2019:i:c:p:214-220
    DOI: 10.1016/j.amc.2018.12.009
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    References listed on IDEAS

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    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. Du, Zhibin, 2017. "Further results regarding the sum of domination number and average eccentricity," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 299-309.
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    Cited by:

    1. Hong Gao & Kun Li & Yuansheng Yang, 2019. "The k -Rainbow Domination Number of C n □ C m," Mathematics, MDPI, vol. 7(12), pages 1-19, December.
    2. Hong Gao & Changqing Xi & Yuansheng Yang, 2020. "The 3-Rainbow Domination Number of the Cartesian Product of Cycles," Mathematics, MDPI, vol. 8(1), pages 1-20, January.

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