Discussiones Mathematicae Graph Theory 31(3) (2011) 533-545
doi: 10.7151/dmgt.1563

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Closed k-stop distance in graphs

Grady Bullington1, Linda Eroh1, Ralucca Gera2 and Steven J. Winters1

1Department of Mathematics
University of Wisconsin Oshkosh
Oshkosh, WI 54901 USA
2Department of Applied Mathematics
Naval Postgraduate School
Monterey, CA 93943 USA


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) =
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.

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) ≤ 2 rad(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.

Keywords: Traveling Salesman, Steiner distance, distance, closed k-stop distance

2010 Mathematics Subject Classification: 05C12, 05C05.


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Received 4 June 2009
Revised 6 August 2010
Accepted 6 August 2010