On the broadcast independence number of caterpillars

Abstract

Let G be a simple undirected graph. A broadcast on G is a function f:V(G)→N such that f(v)≤eG(v) holds for every vertex v of G, where eG(v) denotes the eccentricity of v in G, that is, the maximum distance from v to any other vertex of G. The cost of f is the value cost(f)=∑v∈V(G)f(v). A broadcast f on G is independent if for every two distinct vertices u and v in G, dG(u,v)>max{f(u),f(v)}, where dG(u,v) denotes the distance between u and v in G. The broadcast independence number of G is then defined as the maximum cost of an independent broadcast on G.In this paper, we study independent broadcasts of caterpillars and give an explicit formula for the broadcast independence number of caterpillars having no pair of adjacent trunks, a trunk being an internal spine vertex with degree 2.