Constant Local multiset dimension of Convex Polytopes-$R_n$ and $S_n$

Abstract

In graph theory, the metric dimension is a concept whose primary purpose is to uniquely identify each vertex in a graph based on distance with respect to a collection of reference vertices, called  resolving sets. Resolving sets are used in communication networks to uniquely identify the position of each node. In robotics and navigation systems, robots can determine their exact position with reference to a set of codes generated by resolving set. Based on the practical requirements of the network problems, there are several variation in metric dimensions. Local multiset dimension (LMD) of a graph is one such variation, in which a local multiset basis generates a distinct multiset of distances for every pair of adjacent vertices with reference to a subset of the vertex set of the graph.  Convex polytopes are the two-dimensional geometric shapes that can be used in designing an optimal network configuration. In this paper, we determined the LMD of two convex polytopes and hence found that the local multiset dimension for both the polytopes is constant, irrespective of the order of the graphs.