๐ค AI Summary
This paper investigates the relationship between the multipacking number $ mp(G) $ and the broadcast domination number $ gamma_b(G) $ in graph theory, focusing on cactus graphs and $ delta $-hyperbolic graphs. Using combinatorial constructions, graph decomposition, and hyperbolic geometric analysis, we establish that $ gamma_b(G) leq 1.5,mp(G) + 5.5 $ holds for all cactus graphs $ G $, with the coefficient $ 1.5 $ being tight; we further design an $ O(n) $-time approximation algorithm yielding a solution of size at least $ frac{2}{3}mp(G) - frac{11}{3} $. Crucially, we show that $ gamma_b(G) - mp(G) $ can be arbitrarily large when $ delta = frac{1}{2} $, yet remains bounded for all $ delta < frac{1}{2} $โestablishing $ frac{1}{2} $ as the infimum hyperbolicity constant separating the two parameters unboundedly. Finally, we construct an infinite family of graphs achieving the ratio $ gamma_b / mp = 4/3 $, each attaining hyperbolicity exactly $ frac{1}{2} $.
๐ 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 $gamma_b(G)leq c_1cdot mp(G)+c_2$, 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 $frac{1}{2}$.
