Closed 3-stop center and periphery in graphs

A delivery person must leave the central location of the business, deliver packages at a number of addresses, and then return. Naturally, he/she wishes to reduce costs by finding the most efficient route. This motivates the following Given a set of k distinct vertices S={x1,x2,.,xk} in a simple graph G, the closed k-stop-distance of set S is defined to be dk(S)=min..P(S)(d(.(x1),.(x2))+d(.(x2),.(x3))+.+d(.(xk).(x1))), w here P(S) is the set of all permutations of S. That is the same as saying that d k (S) is the length of a shortest closed walk through the vertices {x 1, ..., x k . The closed 2-stop distance is twice the standard distance between two vertices. We study the closed k-stop center and closed k-stop periphery of a graph, for k = 3.