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 $ γ_{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 $G$, $γ_{b}(G)\leq \big\lceil{\frac{3}{2} mp(G)\big\rceil}$. We also show that $γ_b(G)-mp(G)$ can be arbitrarily large for connected chordal graphs by constructing an infinite family of connected chordal graphs such that the ratio $γ_b(G)/mp(G)=10/9$, with $mp(G)$ arbitrarily large. Moreover, we show that $γ_{b}(G)\leq \big\lfloor{\frac{3}{2} mp(G)+2δ\big\rfloor} $ holds for all $δ$-hyperbolic graphs. In addition, we provide a polynomial-time algorithm to construct a multipacking of a $δ$-hyperbolic graph $G$ of size at least $ \big\lceil{\frac{2mp(G)-4δ}{3} \big\rceil} $.

翻译：对于具有顶点集$V$和边集$E$的图$G=(V,E)$，若函数$f:V\rightarrow\{0,1,2,...,diam(G)\}$满足：对每个顶点$u\in V$，存在$G$中顶点$v$（可能$u=v$）使得$f(v)>0$且$d(u,v)\leq f(v)$，则称$f$为$G$上的支配广播。支配广播$f$的成本为$\sum_{v\in V}f(v)$。支配广播的最小成本称为$G$的广播控制数，记为$\gamma_{b}(G)$。多填装是图$G=(V,E)$中满足以下条件的子集$S\subseteq V$：对任意顶点$v\in V$和任意整数$r\geq 1$，以$v$为中心、半径为$r$的球至多包含$S$的$r$个顶点，即$G$中与$v$距离不超过$r$的$S$中顶点数不超过$r$。$G$的多填装数是其多填装的最大基数，记为$mp(G)$。本文证明：对任意连通弦图$G$，有$\gamma_{b}(G)\leq \big\lceil{\frac{3}{2} mp(G)\big\rceil}$。通过构造无限族连通弦图，我们进一步证明$\gamma_b(G)-mp(G)$在连通弦图上可以任意大，且该图族满足比值$\gamma_b(G)/mp(G)=10/9$而$mp(G)$可任意大。此外，我们证明对所有$\delta$-双曲图均成立$\gamma_{b}(G)\leq \big\lfloor{\frac{3}{2} mp(G)+2\delta\big\rfloor}$。最后，我们给出多项式时间算法，可为$\delta$-双曲图$G$构造基数至少为$\big\lceil{\frac{2mp(G)-4\delta}{3} \big\rceil}$的多填装。