# Notes on k-rainbow independent domination in graphs

## TL;DR

This paper studies the $k$-rainbow independent domination number in graphs, improves bounds on its sum with the complement, and proves the NP-completeness of related decision problems, addressing open questions in the field.

## Contribution

It refines the Nordhaus-Gaddum-type bounds for the $k$-rainbow independent domination number and establishes NP-completeness for the associated decision problem.

## Key findings

- Improved bounds: $5 \,\leq\, \gamma_{rik}(G)+\gamma_{rik}(\overline{G}) \leq n+2$ for non-5-cycle graphs.
- NP-completeness: Deciding the existence of a $k$-rainbow independent dominating function of a given weight is NP-complete.
- Responds to open questions by umenjak et al.

## Abstract

The $k$-rainbow independent domination number of a graph $G$, denoted $\gamma_{\rm rik}(G)$, is the cardinality of a smallest set consisting of two vertex-disjoint independent sets $V_1$ and $V_2$ for which every vertex in $V(G)\setminus (V_1\cup V_2)$ has neighbors in both $V_1$ and $V_2$. This domination invariant was proposed by {\v{S}}umenjak, Rall and Tepeh in (Applied Mathematics and Computation 333(15), 2018: 353-361), which allows to reduce the problem of computing the independent domination number of the generalized prism $G {\Box} K_k$ to an integer labeling problem on $G$. They proved a Nordhaus-Gaddum-type theorem: $5\leq \gamma_{\rm rik}(G)+\gamma_{\rm rik}(\overline{G})\leq n+3$ for every graph $G$ of order $n\geq 3$, where $\overline{G}$ is the complement of $G$. In this paper, we improve this result by showing that if $G$ is not isomorphic to the 5-cycle, then $5\leq \gamma_{\rm rik}(G)+\gamma_{\rm rik}(\overline{G})\leq n+2$. Moreover, we show that the problem of deciding whether a graph has a $k$-rainbow independent dominating function of a given weight is $\mathcal{NP}$-complete. Our results respond some open questions proposed by \v{S}umenjak, et al.