On families of convex polytopes with constant metric dimension

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 = { w 1 , w 2 , … , w k } is called a resolving set for G if for every two distinct vertices x , y ∈ V ( G ) , there is a vertex w i ∈ 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 d i m ( G ) . A family G of connected graphs is a family with constant metric dimension if d i m ( G ) is finite and does not depend upon the choice of G in G . In this paper, we study the metric dimension of some classes of convex polytopes which are obtained by the combinations of two different graph of convex polytopes. It is shown that these classes of convex polytopes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes.