# A simpler PTAS for connected *k*-path vertex cover in homogeneous wireless sensor network

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## Abstract

Because of its application in the field of security in wireless sensor networks, *k*-path vertex cover (\(\hbox {VCP}_k\)) has received a lot of attention in recent years. Given a graph \(G=(V,E)\), a vertex set \(C\subseteq V\) is a *k*-path vertex cover (\(\hbox {VCP}_k\)) of *G* if every path on *k* vertices has at least one vertex in *C*, and *C* is a connected *k*-path vertex cover of *G* (\(\hbox {CVCP}_k\)) if furthermore the subgraph of *G* induced by *C* is connected. A homogeneous wireless sensor network can be modeled as a unit disk graph. This paper presents a new PTAS for \(\hbox {MinCVCP}_k\) on unit disk graphs. Compared with previous PTAS given by Liu et al., our method not only simplifies the algorithm and reduces the time-complexity, but also simplifies the analysis by a large amount.

## Keywords

Connected*k*-path vertex cover Unit disk graph PTAS Approximation algorithm

## Notes

### Acknowledgements

This research is supported by NSFC (11771013, 11531011, 61502431).

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