The broadcast domination–multipacking inequality

For a graph $G=(V,E)$, a multipacking is a set $M\subseteq V$ such that, for every vertex $v\in V$ and every integer $k\geq 1$, $|N_k[v]\cap M|\leq k$, where $N_k[v]$ is the set of vertices at distance at most $k$ from $v$. The multipacking number $\operatorname{mp}(G)$ is the maximum cardinality of a multipacking, and $\gamma_b(G)$ denotes the broadcast domination number of $G$. Beaudou's conjecture. For every graph $G$,

$$\gamma_b(G)\leq 2\operatorname{mp}(G).$$

The conjecture removes the additive constant from the previously known bound $\gamma_b(G)\leq 2\operatorname{mp}(G)+3$. The factor $2$ is necessary for some small graphs, including $C_4$ and $C_5$, for which $\operatorname{mp}(G)=1$ and $\gamma_b(G)=2$, as well as examples with $\operatorname{mp}(G)=2$ and $\gamma_b(G)=4$.