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Closed k-stop distance in graphs

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Bullington G., Eroh L., Gera R., Winters S.

Discussiones Mathematicae Graph Theory

2011

tom 31

nr 3

533-545

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. 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.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Bullington G.

1 Eroh L.

1 Gera R.

1 Winters S.

1 2011

Wyniki wyszukiwania

Closed k-stop distance in graphs