A survey on alliances and related parameters in graphs

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.