Geodesic
The weighted $k$-path vertex cover problem on series-parallel graphs
Trudy Instituta matematiki, Tome 25 (2017) no. 1, pp. 62-81

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Given a graph $G$ with a vertex weight function $\omega_V:~V(G)\to\mathbb{R}^+$ and a positive integer $k,$ we consider the weighted $k$-path vertex cover problem: it consists in finding a minimum-weight subset $S$ of vertices of a graph $G$ such that every path of order $k$ in $G$ contains at least one vertex from $S.$ We give $O(n)$ algorithms for finding the minimum weight of $k$-path vertex cover and connected $k$-path vertex cover for series-parallel graphs. 
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     author = {V. V. Lepin},
     title = {The weighted $k$-path vertex cover problem on series-parallel graphs},
     journal = {Trudy Instituta matematiki},
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     publisher = {mathdoc},
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     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2017_25_1_a5/}
}
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V. V. Lepin. The weighted $k$-path vertex cover problem on series-parallel graphs. Trudy Instituta matematiki, Tome 25 (2017) no. 1, pp. 62-81. http://geodesic.mathdoc.fr/item/TIMB_2017_25_1_a5/