DMGT

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

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

Abstract

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, …,x_k } in a simple graph G, the closed k-stop-distance of set S is defined to be

d_k (S) =

min
Θ ∈ P(S)

⎛
⎜
⎝

d( Θ(x₁), Θ(x₂)) + d( Θ(x₂), Θ(x₃)) + …+ d( Θ(x_k), Θ(x₁))

⎞
⎟
⎠

where P(S) is the set of all permutations from S onto S. That is the same as saying that d_k (S) is the length of the shortest closed walk through the vertices {x₁, …,x_k}. 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 rad_k(G) ≤ diam_k(G) ≤ k/(k −1) rad_k(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.

References

[1]	G. Chartrand and P. Zhang, Introduction to Graph Theory (McGraw-Hill, Kalamazoo, MI, 2004).
[2]	G. Chartrand, O.R. Oellermann, S. Tian and H.-B. Zou, Steiner distance in graphs, Casopis Pro Pestován'i Matematiky 114 (1989) 399--410.
[3]	J. Gadzinski, P. Sanders, and V. Xiong, k-stop-return distances in graphs, unpublished manuscript.
[4]	M.A. Henning, O.R. Oellermann, and H.C. Swart, On Vertices with Maximum Steiner [eccentricity in graphs]. Graph Theory, Combinatorics, Algorithms, and Applications (San Francisco, CA, (1989)). SIAM, Philadelphia, PA (1991), 393--403.
[5]	M.A. Henning, O.R. Oellermann, and H.C. Swart, On the Steiner Radius and Steiner Diameter of a Graph. Ars Combin. 29C (1990) 13--19.
[6]	O.R. Oellermann, On Steiner Centers and Steiner Medians of Graphs, Networks 34 (1999) 258--263, doi: 10.1002/(SICI)1097-0037(199912)34:4<258::AID-NET4>3.0.CO;2-2.

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