On the edge metric dimension of some classes of cacti

The cactus graph has many practical applications, particularly in radio communication systems. Let $G = (V, E)$ be a finite, undirected, and simple connected graph, then the edge metric dimension of $G$ is the minimum cardinality of the edge metric generator for $G$ (an ordered set of vertices that uniquely determines each pair of distinct edges in terms of distance vectors). Given an ordered set of vertices $\mathcal{G}_e = \{g_1, g_2, ..., g_k \}$ of a connected graph $G$, for any edge $e\in E$, we referred to the $k$-vector (ordered $k$-tuple), $r(e|\mathcal{G}_e) = (d(e, g_1), d(e, g_2), ..., d(e, g_k))$ as the edge metric representation of $e$ with respect to $G_e$. In this regard, $\mathcal{G}_e$ is an edge metric generator for $G$ if, and only if, for every pair of distinct edges $e_1, e_2 \in E$ implies $r (e_1 |\mathcal{G}_e) \neq r (e_2 |\mathcal{G}_e)$. In this paper, we investigated another class of cacti different from the cacti studied in previous literature. We determined the edge metric dimension of the following cacti: $\mathfrak{C}(n, c, r)$ and $\mathfrak{C}(n, m, c, r)$ in terms of the number of cycles $(c)$ and the number of paths $(r)$.