Discussiones Mathematicae Graph Theory

Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating 2-broadcasts along with the associated parameter, the dominating 2-broadcast number. We prove that computing the dominating 2-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desired.

This paper presents a comprehensive review of the literature on the original concept of metric location, along with its various adaptations and extensions that have been developed over time. Given that determining a minimum location set is generally NP-hard, we focus on analyzing the behavior of these sets within specific graph families, including paths, cycles, trees and unicyclic graphs. In addition to synthesizing existing knowledge, we contribute new findings and insights to the field, advancing the understanding of metric location problems in these structured graph classes.