Partial Domination in Graphs

TL;DR

This paper introduces the concept of alpha-partial domination in graphs, analyzing bounds, spectrum, and Nordhaus-Gaddum relations for the alpha-partial domination number, which generalizes classical domination concepts.

Contribution

It defines the alpha-partial domination number, explores bounds and spectral properties, and establishes Nordhaus-Gaddum type inequalities, advancing the understanding of partial domination in graphs.

Findings

Bounds on alpha-partial domination number based on graph parameters

Introduction of alpha-partial domination spectrum

Nordhaus-Gaddum bounds for partial domination number

Abstract

Let $G=(V,E)$ be a graph. For some $\alpha$ with $0<\alpha \leq 1$, a subset $S$ of $V$ is said to be a $\alpha$-partial dominating set if $|N[S]|\geq \alpha |V|$. The size of a smallest such $S$ is called the $\alpha$-partial domination number and is denoted by $\mathsf{pd}_\alpha(G)$. In this paper, we introduce $\alpha$-partial domination number in a graph $G$ and study different bounds on the partial domination number of a graph $G$ with respect to its order, maximum degree, domination number etc., Moreover, $\alpha$-partial domination spectrum is introduced and Nordhaus-Gaddum bounds on the partial domination number are studied.