# On three outer-independent domination related parameters in graphs

## TL;DR

This paper investigates three related domination parameters in graphs, providing bounds, complexity results, and Nordhaus-Gaddum inequalities for each, advancing understanding of their properties and computational aspects.

## Contribution

It introduces and analyzes the $2$-outer-independent, total outer-independent, and double outer-independent domination numbers, including bounds, complexity, and inequalities.

## Key findings

- Derived bounds for the domination parameters.
- Established computational complexity results.
- Proved Nordhaus-Gaddum type inequalities.

## Abstract

Let $G$ be a graph and let $S\subseteq V(G)$. The set $S$ is a double outer-independent dominating set of $G$ if $|N[v]\cap D|\geq2$, for all $v\in V(G)$, and $V(G)\setminus S$ is independent. Similarly, $S$ is a $2$-outer-independent dominating set, if every vertex from $V(G)\setminus S$ has at least two neighbors in $S$ and $V(G)\setminus S$ is independent. Finally, $S$ is a total outer-independent dominating set if every vertex from $V(G)$ has a neighbor in $S$ and the complement of $S$ is an independent set. The double, total or $2$-outer-independent domination number of $G$ is the smallest possible cardinality of any double, total or $2$-outer-independent dominating set of $G$, respectively. In this paper, the $2$-outer-independent, the total outer-independent and the double outer-independent domination numbers of graphs are investigated. We prove some Nordhaus-Gaddum type inequalities, derive their computational complexity and present several bounds for them.