# Independent Rainbow Domination of Graphs

Bulletin of the Malaysian Mathematical Sciences Society, Apr 2017

Given a positive integer t and a graph F, the goal is to assign a subset of the color set $\{1,2,\ldots ,t\}$ to every vertex of F such that every vertex with the empty set assigned has all t colors in its neighborhood. Such an assignment is called the t-rainbow dominating function ($t\mathrm{RDF}$) of the graph F. A $t\mathrm{RDF}$ is independent ($It\mathrm{RDF}$) if vertices assigned with non-empty sets are pairwise non-adjacent. The weight of a $t\mathrm{RDF}$ g of a graph F is the value $w(g) =\sum _{v \in V(F)}|g(v)|$. The independent t-rainbow domination number $i_{rt}(F)$ is the minimum weight over all $It\mathrm{RDF}$s of F. In this article, it is proved that the independent t-rainbow domination problem is NP-complete even if the input graph is restricted to a bipartite graph or a planar graph, and the results of the study provide some bounds for the independent t-rainbow domination number of any graph for a positive integer t. Moreover, the exact values and bounds of the independent t-rainbow domination numbers of some Petersen graphs and torus graphs are given.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs40840-017-0488-6.pdf

Zehui Shao, Zepeng Li, Aljoša Peperko, Jiafu Wan, Janez Žerovnik. Independent Rainbow Domination of Graphs, Bulletin of the Malaysian Mathematical Sciences Society, 2017, 1-19, DOI: 10.1007/s40840-017-0488-6