A broadcast on a nontrivial connected graph G with vertex set V is a function f from V to { 0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all v in V. The weight of f is the sum of the function values taken over V. A vertex u hears f from v if f(v) is positive and d(u,v) is at most f(v). A broadcast f is dominating if every vertex of G hears f. The upper broadcast number of G is { \Gamma}_{ b}(G), which is the maximum weight of a minimal dominating broadcast on G. A broadcast f is boundary independent if, for any vertex w that hears f from vertices v_{ 1},...,v_{ k}, where k is at least 2, the distance d(w,v_{ i}) equals f(v_{ i}) for each i. The maximum weight of a boundary independent broadcast is the boundary independence broadcast number { \alpha}_{ bn}(G). We compare { \alpha}_{ bn} to { \Gamma}_{ b}, showing that neither is an upper bound for the other. We show that the differences { \Gamma}_{ b}-{ \alpha}_{ bn} and { \alpha}_{ bn}-{ \Gamma}_{ b} are unbounded, the ratio { \alpha}_{ bn}/{ \Gamma}_{ b} is bounded for all graphs, and { \Gamma}_{ b}/{ \alpha}_{ bn} is bounded for bipartite graphs but unbounded in general.
Categories: math.CO