Geodesic
Algorithms for solution $k$-Star Hub Problem for trees and series-parallel graphs
Trudy Instituta matematiki, Tome 15 (2007) no. 2, pp. 48-57.

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The following problem is considered. The $k$-Star Hub Problem. Input: Given a graph $G=(V,E)$, a nonnegative integer weight function $w_0\colon E\to\mathbb{Z}^+$ on the edges and $k$ nonnegative integer weight functions on the vertices $w_1,\ldots,w_k\colon V\to\mathbb{Z}^+$. Objective: Find a set of edges $F\subseteq E$ and $k$ subsets of the vertices $V_1,\ldots,V_k\subseteq V$ such that for all $e=(u,v)\in E$ either $e\in F$ or for some $i\in\{1,\ldots,k\}$ $\{u,v\}\in V_i$, and $$ \sum_{e\in F}w_0(e)+\sum_{i=1}^k\,\sum_{v\in V_i}w_i(v) $$ is minimal. Linear-time algorithms for this problem when $k$ is fixed and $G$ is a tree or a series-parallel graph are given. 
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V. V. Lepin. Algorithms for solution $k$-Star Hub Problem for trees and series-parallel graphs. Trudy Instituta matematiki, Tome 15 (2007) no. 2, pp. 48-57. http://geodesic.mathdoc.fr/item/TIMB_2007_15_2_a5/

[1] Blasum U., Hochstattler W., Oertel P., Steiner diagrams and $k$-star hubs, Report No. 00.384, Angewandte mathematik und informatik Universität zu Köln, 1999 | MR

[2] Hochstattler W., Oertel P., The 5-Star-Hub-Problem is NP-complete, Report No. 00.385, Angewandte mathematik und informatik Universität zu Köln, 2000

[3] Bern M.W., Lawler E.L., Wong A.L., “Linear time computation of optimal subgraphs of decomposable graphs”, J. Algorithms, 8 (1987), 216–235 | DOI | MR | Zbl

[4] Borie R.B., Parker R.G., Tovey C.A., “Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families”, Algorithmica, 7 (1992), 555–581 | DOI | MR | Zbl

[5] Kikuno T., Yoshida N., Kakuda Y., “A linear algorithm for the domination number of a series-parallel graph”, Disc. Appl. Math., 5 (1983), 299–311 | DOI | MR | Zbl

[6] Takamizawa K., Nishizeki T., Saito N., “Linear-time computability of combinatorial problems on series-parallel graphs”, J. ACM, 29 (1982), 623–641 | DOI | MR | Zbl

[7] Duffin R.J., “Topology of series-parallel graphs”, J. Math. Anal. Appl., 10 (1965), 303–318 | DOI | MR | Zbl

[8] Valdes J., Tarjan R.E., Lawler E.L., “The recognition of series parallel digraphs”, SIAM J. Comput., 11 (1982), 298–313 | DOI | MR | Zbl