Comparing upper broadcast domination and boundary independence broadcast numbers of graphs

A broadcast on a nontrivial connected graph G = (V, E) is a function f : V ->{0, 1, ... , d}, where d = diam(G), such that f(v) <= e(v) (the eccentricity of v) for all v is an element of V. The weight of f is sigma(f) = Sigma(v is an element of V)f(v). A vertex u hears f from v if f(v) > 0 and d(u, v) <= f(v). A broadcast f is dominating if every vertex of G hears f. The upper broadcast domination number of G is Gamma(b)(G) = max {sigma(f) : f is a minimal dominating broadcast of G} .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 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.