Lower boundary independent and hearing independent broadcasts in graphs

A broadcast on a connected graph $G$ with vertex set $V(G)$ is a function $f:V(G)\rightarrow \{0, 1,..., \text{diam}(G)\}$ such that $f(v)\leq e(v)$ (the eccentricity of $v$) for all $v\in V(G)$. A vertex $v$ is said to be broadcasting if $f(v)>0$, with the set of all such vertices denoted $V_f^+$. A vertex $u$ hears $f$ from $v\in V_f^+$ if $d_G(u, v)\leq f(v)$. The broadcast $f$ is hearing independent if no broadcasting vertex hears another. If, in the broadcast $f$, any vertex $u$ that hears $f$ from multiple broadcasting vertices satisfies $f(v)\leq d_G(u, v)$ for all $v\in V_f^+$, it is said to be boundary independent. The cost of $f$ is \( \sigma(f)=\sum_{v\in V(G)}f(v) \). The minimum cost of a maximal boundary independent broadcast on $G$, called the lower bn-independence number, is denoted by $i_{bn}(G)$. The lower h-independence number $i_h(G)$ is defined analogously for hearing independent broadcasts. We prove that for an arbitrary connected graph $G$, either parameter equals the minimum of the corresponding parameter among that of the spanning trees of $G$. We use these results to prove that $i_{bn}(G)\leq i_h(G)$ for all graphs $G$. We also show that $i_h(G)//i_{bn}(G)$ is bounded.