Relation Between Broadcast Domination and Multipacking Numbers on Chordal Graphs

For a graph $$ G = (V, E) with vertex set V and an edge E, function f : \rightarrow \{0, 1, 2, . , diam(G)\} is called broadcast on G. each u \in if there exists v in (possibly, ) such that (v) > 0 d(u, v) \le then dominating The cost of the quantity \sum _{v\in V}f(v) minimum domination number G, denoted by \gamma _{b}(G) A multipacking S \subseteq for every integer r \ge 1 ball radius around contains at most vertices S, is, are distance from maximum cardinality {{\,\textrm{mp}\,}}(G) It known $${{\,\textrm{mp}\,}}(G)\le _b(G)$$ $$\gamma _b(G)\le 2{{\,\textrm{mp}\,}}(G)+3$$ any it was shown _b(G)-{{\,\textrm{mp}\,}}(G)$$ can be arbitrarily large connected graphs (as exist infinitely many where _b(G)/ {{\,\textrm{mp}\,}}(G)=4/3$$ $${{\,\textrm{mp}\,}}(G)$$ large). strongly chordal graphs, $${{\,\textrm{mp}\,}}(G)=\gamma always holds. We show that, _{b}(G)\le \big \lceil {\frac{3}{2} {{\,\textrm{mp}\,}}(G)\big \rceil }$$ also constructing infinite family ratio _b(G)/{{\,\textrm{mp}\,}}(G)=10/9$$ large. This result shows we cannot improve bound to form c_1\cdot {{\,\textrm{mp}\,}}(G)+c_2$$ constant $$c_1<10/9$$ $$c_2$$