On $k$-rainbow independent domination in graphs

TL;DR

This paper introduces the $k$-rainbow independent domination number, a new graph invariant related to independent domination in generalized prisms, providing bounds, exact values, and a Nordhaus-Gaddum-type theorem.

Contribution

It defines the $k$-rainbow independent domination number, establishes bounds and exact values, and proves a Nordhaus-Gaddum-type theorem for the case $k=2$.

Findings

Bounds for the $k$-rainbow independent domination number are established.

Exact values are determined for specific graph classes.

A sharp Nordhaus-Gaddum-type inequality for $k=2$ is proved.

Abstract

In this paper, we define a new domination invariant on a graph $G$, which coincides with the ordinary independent domination number of the generalized prism $G \Box K_k$, called the $k$-rainbow independent domination number and denoted by $\gamma_{{\rm ri}k}(G)$. Some bounds and exact values concerning this domination concept are determined. As a main result, we prove a Nordhaus-Gaddum-type theorem on the sum for $2$-rainbow independent domination number, and show if G is a graph of order $n \geq 3$, then $5\leq \gamma_{{\rm ri}2}(G)+\gamma_{{\rm ri}2}(\overline{G})\leq n+3$, with both bounds being sharp.