On the Broadcast Dimension of a Graph

TL;DR

This paper investigates the broadcast dimension of graphs, providing tight bounds for acyclic graphs, analyzing the effects of edge deletion, and answering key open questions in the field.

Contribution

It derives asymptotically tight bounds on broadcast dimension for acyclic graphs and general graphs, and analyzes how edge deletion impacts broadcast dimension.

Findings

Lower bound on broadcast dimension for acyclic graphs

Asymptotic tightness of broadcast dimension bounds for general graphs

Edge deletion can cause unbounded increase or decrease in broadcast dimension

Abstract

A function $f:V(G)\rightarrow \mathbb{Z}^+ \cup \{0\}$ is a resolving broadcast of a graph $G$ if, for any distinct $x,y\in V(G)$, there exists a vertex $z\in V(G)$ with $f(z)>0$ such that $\min\{d(x,z), f(z)+1\} \neq \min\{d(y,z), f(z)+1\}.$ The broadcast dimension of $G$ is the minimum of $\sum_{v\in V(G)}f(v)$ over all resolving broadcasts $f$ of $G$. The concept of broadcast dimension was introduced by Geneson and Yi as a variant of metric dimension and has applications in areas such as network discovery and robot navigation. In this paper, we derive an asymptotically tight lower bound on the broadcast dimension of an acyclic graph in the number of vertices, and we show that a lower bound by Geneson and Yi on the broadcast dimension of a general graph in the adjacency dimension is asymptotically tight. We also study the change in the broadcast dimension of a graph under a single edge deletion. We show that both the additive increase and decrease of the broadcast dimension of a graph under edge deletion is unbounded. Moreover, we show that under edge deletion, the broadcast dimension of any graph increases by a multiplicative factor of at most 3. These results fully answer three questions asked by Geneson and Yi.