xx ( xxxx ) 1 – 14 2 CLOSED K - STOP DISTANCE IN GRAPHS 3

25 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 2 G. Bullington, R. Gera, L. Eroh And S.J. Winters 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 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 from S onto S. That is the 26 same as saying that dk(S) is the length of the shortest closed walk 27 through the vertices {x1, . . . , xk}. Recall that the Steiner distance 28 sd(S) is the number of edges in a minimum connected subgraph con29 taining all of the vertices of S. We note some relationships between 30 Steiner distance and closed k-stop distance. 31 The closed 2-stop distance is twice the ordinary distance between 32 two vertices. We conjecture that radk(G) ≤ diamk(G) ≤ k k−1radk(G) 33 for any connected graph G for k ≥ 2. For k = 2, this formula reduces 34 to the classical result rad(G) ≤ diam(G) ≤ 2rad(G). We prove the 35 conjecture in the cases when k = 3 and k = 4 for any graph G and for 36 k ≥ 3 when G is a tree. We consider the minimum number of vertices 37 with each possible 3-eccentricity between rad3(G) and diam3(G). We 38 also study the closed k-stop center and closed k-stop periphery of a 39 graph, for k = 3. 40