Lower Boundary Independent Broadcasts in Trees

A broadcast on a connected graph G = (V, E) is a function f : V -> {0, 1, . . ., diam(G)} such that f(v) <= e(v) (the eccentricity of v) for all v is an element of V if |V| >= 2, and f(v) = 1 if V = {v}. The cost of f is sigma (f) = sigma(v)(is an element of)(V )f(v). Let V-f(+) = {v is an element of V : f(v) > 0}. A vertex u hears f from v is an element of V-f(+) if the distance d(u, v) <= f(v). When f is a broadcast such that every vertex x that hears f from more than one vertex in V-f(+) also satisfies d(x, u) >= f(u) for all u is an element of V-f(+), we say that the broadcast only overlaps in boundaries. A broadcast f is boundary independent if it overlaps only in boundaries. Denote by i(bn)(G) the minimum cost of a maximal boundary independent broadcast. We obtain a characterization of maximal boundary independent broadcasts, show that i(bn)(T ') <= i(bn)(T) for any subtree T ' of a tree T, and determine an upper bound for i(bn)(T) in terms of the broadcast domination number of T . We show that this bound is sharp for an infinite class of trees.