Lower Bound and Exact Values for the Boundary Independence Broadcast Number of a Tree

Abstract

A broadcast on a nontrivial connected graph G is a function f from V(G) to the set {0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all vertices v of G. The weight of f is the sum of the function values over V(G). A vertex u hears f from v if f(v) is positive and u is within distance f(v) from v. A broadcast f is boundary independent if any vertex that hears f from two or more vertices is at distance f(v) from each such vertex v. The maximum weight of a boundary independent broadcast on G is denoted by {\alpha}_{bn}(G). We prove a sharp lower bound on {\alpha}_{bn}(T) for a tree T. Combined with a previously determined upper bound, this gives exact values of {\alpha}_{bn}(T) for some classes of trees T. We also determine {\alpha}_{bn}(T) for trees with exactly two branch vertices and use this result to demonstrate the existence of trees for which {\alpha}_{bn} lies strictly between the lower and upper bounds.