Families of plane graphs with constant metric dimension

Let G be a connected graph and d(x,y) be the distance between the vertices x and y. A subset of vertices W = {w ₁,w ₂.,Wk] is called a resolving set for G if for every two distinct vertices x,y 6 V(G), there is a vertex there W such that d(x,w _i) ≠ d(y,W _i). 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). A family Q of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in G- The metric dimension of some classes of plane graphs has been determined in [3], [4], [6], [12], [15] and [19]. We extend this study, by considering a class of plane graphs defined in [1] and two classes of plane graphs which are the convex polytopes defined in [2]. We show that these plane graphs have constant metric dimension and only three vertices appropriately chosen suffice to resolve all the vertices of these classes of plane graphs. It is natural to ask for the characterization of families of plane graphs with constant metric dimension.