Algorithmic aspects of broadcast independence

Abstract

An independent broadcast on a connected graph $G$ is a function $f:V(G)\to {\mathbb{N}}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast independence number ${\alpha }_b(G)$ of $G$ is the largest weight $\sum _{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$. We describe an efficient algorithm that determines the broadcast independence number of a given tree. Furthermore, we show NP-hardness of the broadcast independence number for planar graphs of maximum degree four, and hardness of approximation for general graphs. Our results solve problems posed by Dunbar, Erwin, Haynes, Hedetniemi, and Hedetniemi (2006), Hedetniemi (2006), and Ahmane, Bouchemakh, Sopena (2018).