Lower Boundary Independent Broadcasts in Trees

TL;DR

This paper introduces and characterizes boundary independent broadcasts in trees, providing bounds on their minimal cost and demonstrating the sharpness of these bounds for certain tree classes.

Contribution

It defines boundary independent broadcasts, characterizes maximal boundary independent broadcasts, and establishes bounds related to broadcast domination in trees.

Findings

Characterization of maximal boundary independent broadcasts.

Proved that the boundary independent broadcast number does not increase in subtrees.

Derived an upper bound for the boundary independent broadcast number based on broadcast domination.

Abstract

A broadcast on a connected graph $G=(V,E)$ is a function $f:V\rightarrow \{0,1,\dots,\operatorname{diam}(G)\}$ such that $f(v)\leq e(v)$ (the eccentricity of $v$) for all $v\in V$ if $|V|\geq2$, and $f(v)=1$ if $V=\{v\}$. The cost of $f$ is $\sigma(f)=\sum_{v\in V}f(v)$. Let $V_{f}% ^{+}$ denote the set of vertices $v$ such that $f(v)$ is positive. A vertex $u$ hears $f$ from $v\in V_{f}^{+}$ if the distance $d(u,v)\leq f(v)$. When $f$ is a broadcast such that every vertex $x$ that hears $f$ from more than one vertex in $V_{f}^{+}$ also satisfies $d(x,u)\geq f(u)$ for all $u\in V_{f}^{+}$, we say that the broadcast only overlaps in boundaries. A broadcast $f$ is boundary independent if it overlaps only in boundaries. Denote by $i_{\operatorname{bn}}(G)$ the minimum cost of a maximal boundary independent broadcast. We obtain a characterization of maximal boundary independent broadcasts, show that $i_{\operatorname{bn}}(T^{\prime})\leq i_{\operatorname{bn}}(T)$ for any subtree $T^{\prime}$ of a tree $T$, and determine an upper bound for $i_{\operatorname{bn}}(T)$ in terms of the broadcast domination number of $T$. We show that this bound is sharp for an infinite class of trees.