On the edge metric dimension of convex polytopes and its related graphs

Let G = (V, E) be a connected graph. The distance between the edge e = uv is an element of E and the vertex x is an element of V is given by d(e, x) = min{d(u, x), d(v, x)}. A subset SE of vertices is called an edge metric generator for G if for every two distinct edges e(1), e(2) is an element of E, there exists a vertex x is an element of S-E such that d(e(1), x) not equal d(e(2), x). An edge metric generator containing a minimum number of vertices is called an edge metric basis for G and the cardinality of an edge metric basis is called the edge metric dimension denoted by mu(E) (G). In this paper, we study the edge metric dimension of some classes of plane graphs. It is shown that the edge metric dimension of convex polytope antiprism An, the web graph W-n, and convex polytope D-n are bounded, while the prism related graph D-n* has unbounded edge metric dimension.