Relating broadcast independence and independence

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 $\sum\limits_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$. Clearly, $\alpha_b(G)$ is at least the independence number $\alpha(G)$ for every connected graph $G$. Our main result implies $\alpha_b(G)\leq 4\alpha(G)$. We prove a tight inequality and characterize all extremal graphs

On the Broadcast Independence Number of Caterpillars

International audienceLet $G$ be a simple undirected graph.A broadcast on $G$ isa function $f : V(G)\rightarrow\mathbb{N}$ such that $f(v)\le e_G(v)$ holds for every vertex $v$ of $G$, where $e_G(v)$ denotes the eccentricity of $v$ in $G$, that is, the maximum distance from $v$ to any other vertex of $G$.The cost of $f$ is the value ${\rm cost}(f)=\sum_{v\in V(G)}f(v)$.A broadcast $f$ on $G$ is independent if for every two distinct vertices $u$ and $v$ in $G$, $d_G(u,v)>\max\{f(u),f(v)\}$,where $d_G(u,v)$ denotes the distance between $u$ and $v$ in $G$.The broadcast independence number of $G$ is then defined as the maximum cost of an independent broadcast on $G$. In this paper, we study independent broadcasts of caterpillars and give an explicit formula for the broadcast independence number of caterpillars having no pair of adjacent trunks, a trunk being an internal spine vertex with degree~2