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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 anydistinct $x, y \in V(G)$, 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 fordirected graphs by considering directed distances.  We partially resolve a question of Zhang by obtaining precise bounds for theadjacency 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 ondirected graphs. First, we explicitly calculate the adjacency dimension of adirected complete $k$-ary tree, where every edge is directed towards theleaves. Next, we prove that $\text{adim}(\vec{G}) = \text{bdim}(\vec{G})$ forsome 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$\text{bdim}(\vec{G})$.

Adjacency and Broadcast Dimension of Grid and Directed Graphs