Efficient $k$-limited Dominating Broadcasts in Product Graphs

In a graph $G$, a subset of vertices $S$ is called an efficient dominating set (EDS) if every vertex in the graph is uniquely dominated by exactly one vertex in $S$. A graph is said to be efficiently dominatable if it contains an EDS. Additionally, a function $f: V(G) \rightarrow \{0, 1, 2, \dots, k\}$ is termed a $k$-limited dominating broadcast if, for every vertex $u \in V(G)$, there exists a vertex $v$, with $f(v) \geq 1$ such that $d(u, v) \leq f(v)$. A vertex $u$ is said to be dominated by a vertex $v$. In this work, we unify these two concepts to explore the notion of efficient $k$-limited broadcast domination in graphs. A $k$-limited dominating broadcast $f$ is called an efficient $k$-limited dominating broadcast ($k$-$ELDB$) if each vertex in the graph is dominated exactly once. The minimum value of $k$ for which the given graph $G$ has $k$-$ELDB$ is defined as $mcr(G)$. We prove determining $mcr(G)$ is NP-Complete for general graphs and explore the $mcr(G)$ values and other related parameters on standard graphs and their products.