Comparing Upper Broadcast Domination and Boundary Independence Numbers of Graphs

Abstract: 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.