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 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 same as saying that dk(S) is the length of the shortest closed walk through the vertices {x1, . . . , xk}. Recall that the Steiner distance sd(S) is the number of edges in a minimum connected subgraph containing all of the vertices of S. We note some relationships between Steiner distance and closed k-stop distance. 

• Type: Journal
• Year: 2011
• Where: DMGT
• Authors: Grady Bullington, Linda Eroh, Ralucca Gera, Steven J. Winters
• Keywords: Distinct Vertices, DMGT 2011, Traveling Salesman Problem, X1 X2