On the Broadcast Independence Number of Caterpillars

Let $G$ be a simple undirected graph.A broadcast on $G$ isa function $f : V(G)\rightarrow$\mathbb{N}$$suchthat$f(v)\le e_G(v)$holdsforeveryvertex$v$of$G$,where$e_G(v)$denotestheeccentricityof$v$in$G$, that is, the maximum distance from $v$ to any other vertex of $G$.The cost of $f$ is the value $\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}(f)=\sum \mathrm{\_}v\in V(G)f(v)$.A broadcast $f$ on $G$ is independent if for every two distinct vertices $u$ and $v$ in $G$, $d\mathrm{\_}G(u,v)>\max \{f(u),f(v)\}$,where $d\mathrm{\_}G(u,v)$ 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 vertices with degree 2.