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} $.