Codes from k-resolving sets for stacked prism graphs

Abstract

A k-resolving set S is a set of vertices {v₁, v₂,....v_l} of a graph G(V,E) if for distinct vertices u,w∈V, the lists of distances (dG(u, v₁),dG(u, v₂),...,dG(u, v_l)) and (dG(w, v₁),dG(w, v₂),...,dG(w, v_l)) differ in at least k-positions. The least size of a k-resolving set is called the k-metric basis of G and its cardinality is called the k-metric dimension, denoted by dim_k(G). We determine error correcting codes for stacked prism graphs namely PmCn using k-resolving sets. We have also constructed an infinite family of stacked prisms of k-dimension. In this paper we have studied the k-metric dimension of P_m□C_n. An explicit formula for dim_k(P_m□C_n) is determined and the codes arising from k-resolving sets of P_m□C_n are developed.