A broadcast on a nontrivial connected graph G = (V, E) is a function f :A broadcast f is boundary independent if, for any vertex w that hears f from vertices v 1 , ..., v k , k ≥ 2, the distance d(w, v i ) = f (v i ) for each i. The maximum weight of a boundary independent broadcast is the boundary independence broadcast number α bn (G).We compare α bn to Γ b , showing that neither is an upper bound for the other. We show that the differences Γ b − α bn and α bn − Γ b are unbounded, the ratio α bn /Γ b is bounded for all graphs, and Γ b /α bn is bounded for bipartite graphs but unbounded in general.
A broadcast on a nontrivial connected graph G = (V, E) is a function f :A broadcast f is boundary independent if, for any vertex w that hears f from vertices v 1 , ..., v k , k ≥ 2, we have that d(w, v i ) = f (v i ) for each i. The maximum weight of a boundary independent broadcast on G is denoted by α bn (G). We prove a sharp lower bound on α bn (T ) for a tree T . Combined with a previously determined upper bound, this gives exact values of α bn (T ) for some classes of trees T . We also determine α bn (T ) for trees with exactly two branch vertices and use this result to demonstrate the existence of trees for which α bn lies strictly between the lower and upper bounds.
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