$2$-limited broadcast domination in cubic graphs

For a graph $G$, a function $f:V(G) \to \{0,1,2\}$ is called a $2$-limited dominating broadcast on $G$ if for every vertex $u$, there exists a vertex $v$ such that $f(v)>0$ and the distance between $u$ and $v$ in $G$ is at most $f(v)$. The {\it cost} of $f$ means the value $\sum_{v\in V(G)}f(v)$, and the {\it $2$-limited broadcast domination number} of $G$, denoted by $\gamma_{b,2}(G)$, is the cost of a $2$-limited dominating broadcast on $G$ with minimum cost. Henning, MacGillivray, and Yang (2020) conjectured that $\gamma_{b,2}(G)\leq \frac{|V(G)|}{3}$ for every cubic graph $G$. In this paper, we confirm the conjecture.