Relation between broadcast domination and multipacking numbers on chordal and other hyperbolic graphs

Abstract: 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 $\big\lceil{\frac{2mp(G)-4\delta}{3} \big\rceil}$.