Closed k-stop distance in graphs

The Traveling Salesman Problem (TSP) is still one of the most researched topics in computational mathematics, and we introduce a variant of it, namely the study of the closed k-walks in graphs. We search for a shortest closed route visiting k cities in a non complete graph without weights. This motivates the following definition. Given a set of k distinct vertices = {x₁, x₂, ...,xₖ} in a simple graph G, the closed k-stop-distance of set is defined to be ${\displaystyle dₖ\left(\right)=\underset{\Theta \in \left(\right)}{min}(d(\Theta (x₁),\Theta (x₂))+d(\Theta (x₂),\Theta (x₃))+...+d(\Theta (xₖ),\Theta (x₁)))}$, where () is the set of all permutations from onto . That is the same as saying that dₖ() is the length of the shortest closed walk through the vertices {x₁, ...,xₖ}. Recall that the Steiner distance sd() is the number of edges in a minimum connected subgraph containing all of the vertices of . We note some relationships between Steiner distance and closed k-stop distance. The closed 2-stop distance is twice the ordinary distance between two vertices. We conjecture that radₖ(G) ≤ diamₖ(G) ≤ k/(k -1) radₖ(G) for any connected graph G for k ≤ 2. For k = 2, this formula reduces to the classical result rad(G) ≤ diam(G) ≤ 2rad(G). We prove the conjecture in the cases when k = 3 and k = 4 for any graph G and for k ≤ 3 when G is a tree. We consider the minimum number of vertices with each possible 3-eccentricity between rad₃(G) and diam₃(G). We also study the closed k-stop center and closed k-stop periphery of a graph, for k = 3.