Domination ratio of a family of integer distance digraphs with arbitrary degree

TL;DR

This paper precisely determines the domination ratio of a broad class of integer distance digraphs, linking graph theory with number theory problems like tiling and missing differences.

Contribution

It generalizes previous results by providing an exact domination ratio for graphs with arbitrary degrees, extending understanding of integer distance graphs.

Findings

Exact domination ratio for (,-2,s)"]

Implications for domination numbers of certain circulant graphs

Connections to tiling and missing difference problems in number theory

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 a finite subset $S\subseteq\mathbb{Z}$. The domination ratio of $\Gamma(\mathbb{Z},S)$, defined as the minimum density of its dominating sets, is related to some number theory problems, such as tiling the integers and finding the maximum density of a set of integers with missing differences. We precisely determine the domination ratio of the integer distance graph $\Gamma(\mathbb{Z},\{1,2,\ldots,d-2,s\})$ for any integers $d$ and $s$ satisfying $d\ge2$ and $s\notin[0,d-2]$. Our result generalizes a previous result on the domination ratio of the graph $\Gamma(\mathbb{Z},\{1,s\})$ with $s\in\mathbb{Z}\setminus\{0,1\}$ and also implies the domination number of certain circulant graphs $\Gamma(\mathbb{Z}_n,S)$, where $\mathbb{Z}_n$ is the finite cyclic group of integers modulo $n$ and $S$ is a subset of $\mathbb{Z}_n$.