Edge Metric Dimension of Some Graph Operations

Abstract

Let G = (V, E) be a connected graph. Given a vertex v ∈ V and an edge e = uw ∈ E, the distance between v and e is defined as d G (e, v) = min{d G (u, v), d G (w, v)}. A nonempty set S ⊂ V is an edge metric generator for G if for any two edges e 1 , e 2 ∈ E there is a vertex w ∈ S such that d G (w, e 1 ) = d G (w, e 2 ). The minimum cardinality of any edge metric generator for a graph G is the edge metric dimension of G. The edge metric dimension of the join, lexicographic and corona product of graphs is studied in this article.

A Combinatorial Approach to the Computation of the Fractional Edge Dimension of Graphs

“…This topic is an active field of research for graph theorists, and many research articles have been published on this subject. For details, see [8,9].…”

On Mixed Metric Dimension of Rotationally Symmetric Graphs

“…The minimum cardinality of an edge metric generator of G is known as edge metric dimension of G, and it is represented as dim e (G). Recently, this variant has been investigated by [19], [24], [25]. Now a new type of dimension is introduced by [16], which is a mixed version of both metric and edge metric dimensions, and authors called it a mixed metric dimension.…”

Bounds for metric dimensions of generalized neighborhood corona graphs

“…Several studies then expand the topic to include several variations. Fernau [8] further expand the work to include adjacency, local, and local adjacency metric dimension, while Peterin [9] obtained edge metric dimension for corona graphs. In addition, Rinurwati et al [10] analized metric dimensions for edge-corona graphs.…”