Alliances and related parameters in graphs

TL;DR

This paper unifies various graph parameters related to alliances and domination, revealing their connections across different areas and proposing a comprehensive framework for their study.

Contribution

It introduces a unifying (global) (D,O)-alliances framework that encompasses known parameters and facilitates their analysis and comparison.

Findings

Identifies connections between multiple graph parameters across different fields.

Proposes a new comprehensive framework for alliances and domination.

Provides a survey highlighting how results transfer among these parameters.

Abstract

In this paper, we show that several graph parameters are known in different areas under completely different names. More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence, positive influence domination, and a parameter associated to fast information propagation in networks to parameters related to various notions of global $r$-alliances in graphs. We also propose a new framework, called (global) $(D,O)$-alliances, not only in order to characterize various known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination. Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations.