Dominating 2-broadcast in graphs: complexity, bounds and extremal graphs

TL;DR

This paper studies the computational complexity and bounds of dominating 2-broadcasts in graphs, proving NP-completeness in general but linear-time solvability for trees, and establishing tight bounds for the parameter.

Contribution

It introduces the dominating 2-broadcast concept, proves NP-completeness for general graphs, provides a linear-time algorithm for trees, and establishes tight bounds for the parameter.

Findings

NP-complete for general graphs

Linear-time algorithm for trees

Tight upper bounds for the dominating 2-broadcast number

Abstract

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.