Multipacking and broadcast domination on cactus graph and its impact on hyperbolic graph

Abstract: 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 $\frac{1}{2}$.