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

Simon Brezovnik and Tadeja Kraner Šumenjak

Applied Mathematics and Computation, 2019, vol. 349, issue C, 214-220

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.

Keywords: Complexity; Algorithm; NP-completeness; Domination; Lexicographic product (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:349:y:2019:i:c:p:214-220

DOI: 10.1016/j.amc.2018.12.009

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