Closed k-stop distance in graphs
Grady Bullington ; Linda Eroh ; Ralucca Gera ; Steven J. Winters
Discussiones Mathematicae Graph Theory, Tome 31 (2011), p. 533-545 / Harvested from The Polish Digital Mathematics Library
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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 dₖ()=minΘ∈()(d(Θ(x₁),Θ(x₂))+d(Θ(x₂),Θ(x₃))+...+d(Θ(xₖ),Θ(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.

Publié le : 2011-01-01
EUDML-ID : urn:eudml:doc:270950 
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Grady Bullington; Linda Eroh; Ralucca Gera; Steven J. Winters. Closed k-stop distance in graphs. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 533-545. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1563/

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