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From Steiner centers to Steiner medians. (English) Zbl 0834.05024

For each subset \(S\) of the vertices of a tree let the Steiner distance \(d(S)\) be the total length of a Steiner subtree connecting \(S\). The Steiner \(n\)-median (\(n\)-center, \((n, k)\)-center) is the set of all vertices \(v\) minimizing the sum (maximum, maximum sum of \(k\) out) of the \(d(S)\) for all \(S\) containing \(v\) and having \(n\) elements. It is shown that the Steiner \(n\)-median and \(n\)-center may be arbitrarily far apart, while the path connecting them consists exactly of all Steiner \((n, k)\)- centers for \(1\leq k\leq (\begin{smallmatrix} p-1\\ n-1\end{smallmatrix})\).

MSC:

05C12 Distance in graphs
05C05 Trees
05C85 Graph algorithms (graph-theoretic aspects)
90B80 Discrete location and assignment
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References:

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