Universal perspectives on irredundance for X-set parameters

Abstract

Universal definitions of irredundance for X-set parameters are presented using blocking sets. This approach is modeled on (domination) irredundance (which uses closed neighborhoods as blocking sets) and zero forcing irredundance (which uses forts). Results include a chain of inequalities between irredundance parameters and original parameters and the isomorphism theorem for TAR reconfiguration graphs of many irredundance parameters. These results are then applied to PSD forcing irredundance and vertex cover irredundance; the chain of inequalities also applies to skew forcing irredundance. The upper vertex cover irredundance number becomes part of the Domination Chain used in the study of (domination) irredundance. Based on the propagation processes involved in forcing, an alternate universal theory of irredundance is developed using closure operators.