Relating broadcast independence and independence

An independent broadcast on a connected graph $G$ is a function $f:V\left(G\right)\to {\mathbb{N}}_0$ such that, for every vertex $x$ of $G$, the value $f\left(x\right)$ is at most the eccentricity of $x$ in $G$, and $f\left(x\right)>0$ implies that $f\left(y\right)=0$ for every vertex $y$ of $G$ within distance at most $f\left(x\right)$ from $x$. The broadcast independence number ${\alpha }_b\left(G\right)$ of $G$ is the largest weight $\sum _{x\in V\left(G\right)}f\left(x\right)$ of an independent broadcast $f$ on $G$. Clearly, ${\alpha }_b\left(G\right)$ is at least the independence number $\alpha \left(G\right)$ for every connected graph $G$. Our main result implies ${\alpha }_b\left(G\right)\le 4\alpha \left(G\right)$. We prove a tight inequality and characterize all extremal graphs.