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

TL;DR

This work investigates the relationship between broadcast domination and multipacking in chordal and $\delta$-hyperbolic graphs. It proves a tight-appearing bound for connected chordal graphs: $\gamma_{b}(G)\leq \lceil\frac{3}{2}\mathrm{mp}(G)\rceil$, while also showing that the gap $\gamma_{b}(G)-\mathrm{mp}(G)$ can be arbitrarily large via a constructive family of chordal graphs with $mp(H_k)=9k$ and $\gamma_b(H_k)=10k$. For $\delta$-hyperbolic graphs, the bound generalizes to $\gamma_{b}(G)\leq \lfloor\frac{3}{2}\mathrm{mp}(G)+2\delta\rfloor$, and the paper provides a polynomial-time method to obtain multipackings of substantial size, specifically at least $\lceil\frac{2\mathrm{mp}(G)-4\delta}{3}\rceil$. The results yield an $(\frac{3}{2}+o(1))$-approximation for multipacking on chordal graphs and establish a framework linking these dual problems across graph classes, with open questions on tightening the universal ratio toward 2 and extending to other graph families.

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 $ 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} $.

Figures (8)

• Figure 1: Inclusion diagram for graph classes mentioned in this paper (and related ones). If a class $A$ has an upward path to class $B$, then $A$ is included in $B$. For the graphs in the gray classes, the broadcast domination number is equal to the multipacking number, but this is not true for the white classes.
• Figure 2: $S_3$ is a connected chordal graph with $\gamma_b(S_3)=2$ and $\mathop{\mathrm{mp}}\nolimits(S_3)=1$
• Figure 3: $F$ is a connected chordal graph with $\gamma_b(F)=3$ and $\mathop{\mathrm{mp}}\nolimits(F)=2$
• Figure 4: $H$ is a connected chordal graph with $\gamma_b(H)=6$ and $\mathop{\mathrm{mp}}\nolimits(H)=4$
• Figure 5: $G_1$ is a connected chordal graph with $\gamma_b(G_1)=5$ and $\mathop{\mathrm{mp}}\nolimits(G_1)=5$. $M_1=\{m_i:1\leq i \leq 5\}$ is a multipacking of size $5$.

Theorems & Definitions (43)

• Proposition 1
• Proposition 2
• Theorem 1
• Corollary 1
• Corollary 2
• Proposition 3
• Proposition 4
• Corollary 3
• Theorem 2: hartnell2014difference
• Theorem 3: erwin2001costteshima2012broadcasts