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.
Cáceres, José, Hernando, M., Mora, M., Pelayo, I. M., Puertas, M. L. Dominating 2-broadcast in graphs: complexity, bounds and extremal graphs. "Applicable analysis and discrete mathematics", 20 Abril 2018, vol. 12, núm. 1, p. 205-223.