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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 $v\in V$. A vertex $v$ is said to bebroadcasting if $f(v)>0$, with the set of all such vertices denoted $V_f^+$. Avertex $u$ 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, inaddition, any vertex $u$ that hears $f$ from multiple broadcasting verticessatisfies $f(v)\leq d_G(u, v)$ for all $v\in V_f^+$, the broadcast is said tobe boundary independent.  The cost of $f$ is $\sigma(f)=\sum_{v\in V(G)}f(v)$. The minimum cost of amaximal boundary independent broadcast on G, called the lower bn-independencenumber, is denoted $i_{bn}(G)$. The lower h-independence number $i_h(G)$ isdefined analogously for hearing independent broadcasts. We prove that$i_{bn}(G)\leq i_h(G)$ for all G and show that $i_h(G)/i_{bn}(G)$ is bounded.For both parameters, we show that the lower bn-independence number(h-independence number) of a connected graph G equals the minimum lowerbn-independence number (h-independence number) among those of its spanningtrees.  We further study the maximum cost of boundary independent broadcasts, denoted$\alpha_{bn}(G)$. We show $\alpha_{bn}(G)$ can be bounded in terms of theindependence number $\alpha(G)$, and prove that the maximum bn-independentbroadcast problem is NP-hard by a reduction from the independent set problem toan instance of the maximum bn-independent broadcast problem.  With particular interest in caterpillars, we investigate bounds on$\alpha_{bn}(T)$ when T is a tree in terms of its order and the number ofvertices of degree at least 3, known as the branch vertices of T. We concludeby describing a polynomial-time algorithm to determine $\alpha_{bn}(T)$ for agiven tree T.

Boundary and Hearing Independent Broadcasts in Graphs and Trees