Boundary independent broadcasts in graphs

Abstract

A broadcast on a nontrivial connected graph \( G = (V,E) \) is a function \( f : V \to \{0,1,\dots,\operatorname{diam}(G)\} \) such that \( f(v) \leq e(v) \) (the eccentricity of \( v \)) for all \( v \in V \). The weight of \( f \) is \( \sigma(f) = \sum_{v \in V} f(v) \). A vertex \( u \) hears \( f \) from \( v \) if \( f(v) > 0 \) and \( d(u,v) \leq f(v) \). A broadcast \( f \) is independent, or hearing independent, if no vertex \( u \) with \( f(u) > 0 \) hears \( f \) from any other vertex \( v \). We define a different type of independent broadcast, namely a boundary independent broadcast, as a broadcast \( f \) such that, if a vertex \( w \) hears \( f \) from vertices \( v_1, \dots, v_k \), \( k \geq 2 \), then \( d(w,v_i) = f(v_i) \) for each \( i \). The maximum weights of a hearing independent broadcast and a boundary independent broadcast are the \textit{hearing independence broadcast number} \( \alpha_h(G) \) and the boundary independence broadcast number \( \alpha_{bn}(G) \), respectively.

We prove that \( \alpha_{bn}(G) = \alpha(G) \) (the independence number) for any 2-connected bipartite graph \( G \) and that \( \alpha_{bn}(G) \leq n – 1 \) for all graphs \( G \) of order \( n \), characterizing graphs for which equality holds. We compare \( \alpha_{bn} \) and \( \alpha_h \) and prove that although the difference \( \alpha_h – \alpha_{bn} \) can be arbitrary, the ratio is bounded, namely \( \alpha_h / \alpha_{bn} < 2 \), which is asymptotically best possible. We deduce that \( \alpha_h(G) \leq 2n – 5 \) for all connected graphs \( G \neq P_n \) of order \( n \), which improves an existing upper bound for \( \alpha_h(G) \) when \( \alpha(G) \geq n/2 \).