For a graph $G$, $ mp(G) $ is the multipacking number, and $\gamma_b(G)$ is the broadcast domination number. It is known that $mp(G)\leq \gamma_b(G)$ and $\gamma_b(G)\leq 2mp(G)+3$ for any graph $G$, and it was shown that $\gamma_b(G)-mp(G)$ can be arbitrarily large for connected graphs. It is conjectured that $\gamma_b(G)\leq 2mp(G)$ for any general graph $G$. We show that, for any cactus graph $G$, $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$. We also show that $\gamma_b(G)-mp(G)$ can be arbitrarily large for cactus graphs and asteroidal triple-free graphs by constructing an infinite family of cactus graphs which are also asteroidal triple-free graphs such that the ratio $\gamma_b(G)/mp(G)=4/3$, with $mp(G)$ arbitrarily large. This result shows that, for cactus graphs, the bound $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$ cannot be improved to a bound in the form $\gamma_b(G)\leq c_1\cdot mp(G)+c_2$, for any constant $c_1<4/3$ and $c_2$. Moreover, we provide an $O(n)$-time algorithm to construct a multipacking of cactus graph $G$ of size at least $ \frac{2}{3}mp(G)-\frac{11}{3} $, where $n$ is the number of vertices of the graph $G$. The hyperbolicity of the cactus graph class is unbounded. For $0$-hyperbolic graphs, $mp(G)=\gamma_b(G)$. Moreover, $mp(G)=\gamma_b(G)$ holds for the strongly chordal graphs which is a subclass of $\frac{1}{2}$-hyperbolic graphs. Now it's a natural question: what is the minimum value of $\delta$, for which we can say that the difference $ \gamma_{b}(G) - mp(G) $ can be arbitrarily large for $\delta$-hyperbolic graphs? We show that the minimum value of $\delta$ is $\frac{1}{2}$ using a construction of an infinite family of cactus graphs with hyperbolicity $\frac{1}{2}$.

翻译：对于图 $G$，$ mp(G) $ 表示多包装数，$\gamma_b(G)$ 表示广播支配数。已知对任意图 $G$ 有 $mp(G)\leq \gamma_b(G)$ 且 $\gamma_b(G)\leq 2mp(G)+3$，同时已证明连通图中 $\gamma_b(G)-mp(G)$ 可以任意大。猜想对任意一般图 $G$ 有 $\gamma_b(G)\leq 2mp(G)$。我们证明，对任意仙人掌图 $G$，有 $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$。通过构造一个无穷仙人掌图族（同时也是无星三元组图），使得比值 $\gamma_b(G)/mp(G)=4/3$，且 $mp(G)$ 任意大，我们证明了在仙人掌图和无星三元组图中 $\gamma_b(G)-mp(G)$ 可以任意大。该结果表明，对仙人掌图而言，不存在常数 $c_1<4/3$ 和 $c_2$ 使得界限 $\gamma_b(G)\leq \frac{3}{2}mp(G)+\frac{11}{2}$ 可改进为 $\gamma_b(G)\leq c_1\cdot mp(G)+c_2$ 的形式。此外，我们提供了一种 $O(n)$ 时间的算法，用于构造仙人掌图 $G$ 中大小至少为 $\frac{2}{3}mp(G)-\frac{11}{3}$ 的多包装，其中 $n$ 为图 $G$ 的顶点数。仙人掌图类的双曲性是无界的。对于 $0$-双曲图，有 $mp(G)=\gamma_b(G)$。此外，对于作为 $\frac{1}{2}$-双曲图子类的强弦图，$mp(G)=\gamma_b(G)$ 同样成立。现在自然的问题是：对于 $\delta$-双曲图，能使得差值 $\gamma_{b}(G) - mp(G)$ 任意大的最小 $\delta$ 值是多少？通过构造一个双曲性为 $\frac{1}{2}$ 的无穷仙人掌图族，我们证明最小 $\delta$ 值为 $\frac{1}{2}$。