Domination Parameters of Graph Covers

TL;DR

This paper explores the relationship between domination parameters and graph covers, establishing bounds and proposing a conjecture on the domination number in such graphs.

Contribution

It introduces the first study linking domination parameters with graph covers, providing bounds and a new conjecture on the domination number.

Findings

Established bounds for domination parameters of graph covers.

Proposed a conjecture on the lower bound of the domination number.

Provided evidence supporting the conjecture.

Abstract

A graph $G$ is a \emph{cover} of a graph $F$ if there exists an onto mapping $\pi : V(G) \to V(F)$, called a (\emph{covering}) \emph{projection}, such that $\pi$ maps the neighbours of any vertex $v$ in $G$ bijectively onto the neighbours of $\pi(v)$ in $F$. This paper is the first attempt to study the connection between domination parameters and graph covers. We focus on the domination number, the total domination number, and the connected domination number. We prove upper and lower bounds for the domination parameters of $G$. Moreover, we propose a conjecture on the lower bound for the domination number of $G$ and provide evidence to support the conjecture.