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 Ge = {g1, g2, ..., gk} of a connected graph G, for any edge e ϵ E, we referred to the k-vector (ordered k-tuple), r(e|Ge) = (d(e, g1), d(e, g2), ..., d(e, gk)) as the edge metric representation of e with respect to Ge. In this regard, Ge is an edge metric generator for G if, and only if, for every pair of distinct edges e1, e2 ϵ E implies r(e1|Ge ) ≠ r(e2|Ge). 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: (Formula Presented)(n, c, r) and (Formula Presented)(n, m, c, r) in terms of the number of cycles (c) and the number of paths (r).