Construction and Properties of Legally Failed 2-Distance Zero Forcing in Some Graph Families.

Source abstract

In this study, we introduce a new variant of failed 2-distance zero forcing in graphs, called legally failed 2-distance zero forcing. The concept extends the framework of 2-distance zero forcing by imposing additional legality conditions on the forcing process, particularly when a forcing set fails to color the entire graph. We define legally failed 2-distance zero forcing sets and the corresponding graph parameter, and investigate their fundamental properties and relationships with existing failed 2-distance zero forcing parameters.