For a graph $ G = (V, E) $ with a vertex set $ V $ and an edge set $ E $, a function $ f : V \rightarrow \{0, 1, 2, . . . , diam(G)\} $ is called a \emph{broadcast} on $ G $. For each vertex $ u \in V $, if there exists a vertex $ v $ in $ G $ (possibly, $ u = v $) such that $ f (v) > 0 $ and $ d(u, v) \leq f (v) $, then $ f $ is called a dominating broadcast on $ G $. The cost of the dominating broadcast $f$ is the quantity $ \sum_{v\in V}f(v) $. The minimum cost of a dominating broadcast is the broadcast domination number of $G$, denoted by $ \gamma_{b}(G) $. A multipacking is a set $ S \subseteq V $ in a graph $ G = (V, E) $ such that for every vertex $ v \in V $ and for every integer $ r \geq 1 $, the ball of radius $ r $ around $ v $ contains at most $ r $ vertices of $ S $, that is, there are at most $ r $ vertices in $ S $ at a distance at most $ r $ from $ v $ in $ G $. The multipacking number of $ G $ is the maximum cardinality of a multipacking of $ G $ and is denoted by $ mp(G) $. We show that, for any connected chordal graph $G$, $\gamma_{b}(G)\leq \big\lceil{\frac{3}{2} mp(G)\big\rceil}$. We also show that $\gamma_b(G)-mp(G)$ can be arbitrarily large for connected chordal graphs by constructing an infinite family of connected chordal graphs such that the ratio $\gamma_b(G)/mp(G)=10/9$, with $mp(G)$ arbitrarily large. Moreover, we show that $\gamma_{b}(G)\leq \big\lfloor{\frac{3}{2} mp(G)+2\delta\big\rfloor} $ holds for all $\delta$-hyperbolic graphs. In addition, we provide a polynomial-time algorithm to construct a multipacking of a $\delta$-hyperbolic graph $G$ of size at least $ \big\lceil{\frac{2mp(G)-4\delta}{3} \big\rceil} $.

翻译：对于图 $G = (V, E)$，其中 $V$ 为顶点集，$E$ 为边集，函数 $f : V \rightarrow \{0, 1, 2, . . . , diam(G)\}$ 称为 $G$ 上的一个**广播**。若对每个顶点 $u \in V$，存在 $G$ 中的某个顶点 $v$（可能 $u = v$）使得 $f(v) > 0$ 且 $d(u, v) \leq f(v)$，则称 $f$ 为 $G$ 上的一个控制广播。控制广播 $f$ 的成本为 $\sum_{v\in V}f(v)$。最小成本的控制广播称为 $G$ 的广播控制数，记为 $\gamma_{b}(G)$。多重打包是图 $G = (V, E)$ 中的一个子集 $S \subseteq V$，使得对每个顶点 $v \in V$ 和每个整数 $r \geq 1$，以 $v$ 为中心半径为 $r$ 的球内至多包含 $S$ 中的 $r$ 个顶点，即 $G$ 中与 $v$ 距离不超过 $r$ 的 $S$ 中顶点数至多为 $r$。$G$ 的多重打包数是 $G$ 中最大多重打包的基数，记为 $mp(G)$。我们证明：对任意连通弦图 $G$，有 $\gamma_{b}(G)\leq \big\lceil{\frac{3}{2} mp(G)\big\rceil}$。同时，通过构造一族无穷连通弦图族，使得比值 $\gamma_b(G)/mp(G)=10/9$ 且 $mp(G)$ 可任意大，我们证明了连通弦图中 $\gamma_b(G)-mp(G)$ 可以任意大。此外，我们证明对所有 $\delta$-双曲图，有 $\gamma_{b}(G)\leq \big\lfloor{\frac{3}{2} mp(G)+2\delta\big\rfloor}$ 成立。最后，我们给出一个多项式时间算法，用于构造 $\delta$-双曲图 $G$ 中规模至少为 $\big\lceil{\frac{2mp(G)-4\delta}{3} \big\rceil}$ 的多重打包。