# Double domination and total $2$-domination in digraphs and their dual problems

## TL;DR

This paper explores double domination and total 2-domination in directed graphs, extending classical graph concepts to digraphs and examining related packing problems.

## Contribution

It introduces and analyzes the concepts of double domination and total 2-domination in digraphs, extending existing graph theories to directed graphs and exploring their dual problems.

## Key findings

- Defined double domination and total 2-domination in digraphs
- Established bounds and properties of these parameters
- Explored relationships with 2-limited packing concepts

## Abstract

A subset $S$ of vertices of a digraph $D$ is a double dominating set (total $2$-dominating set) if every vertex not in $S$ is adjacent from at least two vertices in $S$, and every vertex in $S$ is adjacent from at least one vertex in $S$ (the subdigraph induced by $S$ has no isolated vertices). The double domination number (total $2$-domination number) of a digraph $D$ is the minimum cardinality of a double dominating set (total $2$-dominating set) in $D$. In this work, we investigate these concepts which can be considered as two extensions of double domination in graphs to digraphs, along with the concepts $2$-limited packing and total $2$-limited packing which have close relationships with the above-mentioned concepts.