On metric dimension of flower graphs fn×m and convex polytopes

Abstract

Let G be a connected graph and d(x, y) be the distance between the vertices x and y. A subset of vertices W = {w1,w2, ⋯ , w_k} is called a resolving set for G if for every two distinct vertices x, y ∈ V(G), there is a vertex wi ∈ W such that d(x,wi) ≠ d(y,wi). A resolving set containing a minimum number of vertices is called a metric basis for G and the number of vertices in a metric basis is its metric dimension dim(G). Let F be a family of connected graphs G_n : F = (G_n)_n≥1 depending on n as follows: the order |V(G)| = φ(n) and lim _n→∞φ(n) = ∞. If there exists a constant C > 0 such that dim(G) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension; otherwise F has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), then F is called a family with constant metric dimension. The metric dimension of some classes of plane graphs has been determined in [3-5, 10, 12, 15] and [22], while metric dimension of some classes of convex polytopes has been studied in [10]. In this paper this study is extended, by considering flower graphs Fn×m and two classes of graphs associated to convex polytopes.