# Domination ratio of integer distance digraphs

## TL;DR

This paper investigates the domination ratio in integer distance digraphs, providing foundational results and exact values for specific cases where the generating set has two elements with divisibility relations.

## Contribution

It introduces basic properties of the domination ratio in integer distance digraphs and determines exact ratios for sets of size two with divisibility conditions.

## Key findings

- Established fundamental results on domination ratios.
- Precisely calculated domination ratios for sets with two elements where one divides the other.

## Abstract

An integer distance digraph is the Cayley graph $\Gamma(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to some finite subset $S \subseteq \mathbb{Z}$. The domination ratio of $\Gamma(\mathbb{Z},S)$ is the minimum density of a dominating set in $\Gamma(\mathbb{Z},S)$. We establish some basic results on the domination ratio of $\Gamma(\mathbb{Z},S)$ and precisely determine it when $S=\{s,t\}$ with $s$ dividing $t$.