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Closed 3-stop center and periphery in graphs

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Abstract

A delivery person must leave the central location of the business, deliver packages at a number of addresses, and then return. Naturally, he/she wishes to reduce costs by finding the most efficient route. This motivates the following

Given a set of k distinct vertices \(\mathcal{S} = \{ x_1 ,x_2 , \ldots ,x_k \}\) in a simple graph G, the closed k-stop-distance of set \(\mathcal{S}\) is defined to be

$$d_k (\mathcal{S}) = \mathop {\min }\limits_{\theta \in \mathcal{P}(\mathcal{S})} (d(\theta (x_1 ),\theta (x_2 )) + d(\theta (x_2 ),\theta (x_3 )) + \cdots + d(\theta (x_k )\theta (x_1 ))),$$

where \(\mathcal{P}(\mathcal{S})\) is the set of all permutations of S. That is the same as saying that d k (S) is the length of a shortest closed walk through the vertices {x 1, ..., x k .

The closed 2-stop distance is twice the standard distance between two vertices. We study the closed k-stop center and closed k-stop periphery of a graph, for k = 3.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin Oshkosh, Oshkosh, WI, 54901, USA

    Linda Eroh

  2. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943, USA

    Ralucca Gera

  3. Department of Mathematics, University of Wisconsin Oshkosh, Oshkosh, WI, 54901, USA

    Steven J. Winters

Authors
  1. Linda Eroh
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  2. Ralucca Gera
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  3. Steven J. Winters
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Correspondence to Ralucca Gera.

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Eroh, L., Gera, R. & Winters, S.J. Closed 3-stop center and periphery in graphs. Acta. Math. Sin.-English Ser. 28, 439–452 (2012). https://doi.org/10.1007/s10114-011-0187-4

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  • DOI: https://doi.org/10.1007/s10114-011-0187-4

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