Partial Domination in Graphs

TL;DR

This paper introduces the concept of partial domination in graphs, focusing on sets that dominate at least half of the vertices, relaxing the traditional domination requirement to explore new theoretical properties.

Contribution

It proposes the study of partial domination, specifically 1/2 domination, extending classical domination theory to new fractional domination parameters.

Findings

Defined the 1/2 domination concept in graphs.

Analyzed properties and bounds of partial domination numbers.

Explored implications for domination theory and graph classes.

Abstract

A set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $\gamma(G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a set $S$ is a $\gamma$-set. The single greatest focus of research in domination theory is the determination of the value of $\gamma(G)$. By definition, all vertices must be dominated by a $\gamma$-set. In this paper we propose relaxing this requirement, by seeking sets of vertices that dominate a prescribed fraction of the vertices of a graph. We focus particular attention on $1/2$ domination, that is, sets of vertices that dominate at least half of the vertices of a graph $G$. Keywords: partial domination, dominating set, partial domination number, domination number