Adjacency and Broadcast Dimension of Grid and Directed Graphs

Abstract

Let $G$ be a simple undirected graph. A function $f : V(G) \to \mathbb{Z}_{\geq 0}$ is a $\textit{resolving broadcast}$ of $G$ if for any distinct $x, y \in V(G)$, there exists a vertex $z \in V(G)$ with $f(z) > 0$ such that $\min \{ d(z, x), f(z)+1 \} \neq \min \{ d(z, y), f(z)+1 \}$. The $\textit{broadcast dimension}$ $\text{bdim}(G)$ of $G$ is the minimum of $\sum_{v \in V(G)} f(v)$ over all resolving broadcasts $f$ of $G$. Similarly, the $\textit{adjacency dimension}$ $\text{adim}(G)$ of $G$ is the minimum of $\sum_{v \in V(G)} f(v)$ over all resolving broadcasts $f$ of $G$ where $f$ takes values in $\{0,1\}$. These parameters are defined analogously for directed graphs by considering directed distances. We partially resolve a question of Zhang by obtaining precise bounds for the adjacency dimension of certain Cartesian products of path graphs, namely $\text{adim}(P_2 \square P_n)$ and $\text{adim}(P_3 \square P_n)$. Additionally, we study the behavior of adjacency and broadcast dimension on directed graphs. First, we explicitly calculate the adjacency dimension of a directed complete $k$-ary tree, where every edge is directed towards the leaves. Next, we prove that $\text{adim}(\vec{G}) = \text{bdim}(\vec{G})$ for some particular directed trees $\vec{G}$. Furthermore, we show that $\text{bdim}(G)$ can be as large as an exponential function of $\text{bdim}(\vec{G})$ or as small as a logarithmic function of $\text{bdim}(\vec{G})$.