On the edge metric dimension of some classes of cacti

Abstract

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) $.

Keywords

• cactus graphs
• edge metric generator
• edge metric dimension