Variety of Rational Resolving Sets of Power of a Cycle

Abstract

To discover the actual route and to determine the position of a vertex in the network, we need to select the landmarks by making certain local measurement at the smallest subsets of the nodes. Since each of these measurements are potentially quite costly, the objective here is to minimize the number of measurements which still discover the whole graph. A subset  of vertices of a graph  is called a rational resolving set of  if for each pair  there is a vertex  such that , where  denotes the mean of the distances from the vertex s to all those  A rational resolving set denoted by  set, having minimum  cardinality is a   rational  metric  basis  and its cardinality is the lower    number,  denoted  by .  The   maximum   cardinality  of  a  minimal  set  is  called  the upper    number of , denoted by . In this paper varieties of minimal rational resolving sets of a graph  are defined on the basis of its compliments, called the lower and upper , , ,  numbers and discussed their optimality in power of a cycle