On General Frameworks and Threshold Functions for Multiple Domination

TL;DR

This paper introduces unified frameworks for multiple domination in graphs, generalizing existing concepts, and derives new upper bounds for domination numbers under various restrictions, including degree thresholds.

Contribution

It proposes <r,s>-domination and parametric domination frameworks, extending classical domination concepts and providing new bounds under degree restrictions.

Findings

Generalized upper bounds for multiple domination numbers.

Extended bounds for k-domination and total k-domination.

New bounds considering degree threshold functions.

Abstract

We consider two general frameworks for multiple domination, which are called <r,s>-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the <r,s>-domination and parametric domination numbers. These generalisations imply new upper bounds for the {k}-domination and total k-domination numbers. Also, we study threshold functions, which impose additional restrictions on the minimum vertex degree, and present new upper bounds for the aforementioned numbers. Those bounds extend similar known results for k-tuple domination and total k-domination.