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 θ∈P(S) (d(θ(x1), θ(x2)) + d(θ(x2), θ(x3)) + · · ·+ d(θ(xk), θ(x1))), where P(S) is the set of all permutations of S. That is the same as saying that dk(S) is the length of a shortest closed walk through the vertices {x1, . . . , xk}. 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.