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Approximation algorithms for minimum (weight) connected $$k$$-path vertex cover. (English) Zbl 1333.05168
Summary: A vertex subset $$C$$ of a connected graph $$G$$ is called a connected $$k$$-path vertex cover ($$\mathrm{CVCP}_k$$) if every path on $$k$$ vertices contains at least one vertex from $$C$$, and the subgraph of $$G$$ induced by $$C$$ is connected. This concept originated in the field of security and supervisory control. This paper studies the minimum (weight) $$\mathrm{CVCP}_k$$ problem. We first show that the minimum weight $$\mathrm{CVCP}_k$$ problem can be solved in time $$O(n)$$ when the graph is a tree, and can be solved in time $$O(r n)$$ when the graph is a uni-cyclic graph whose unique cycle has length $$r$$, where $$n$$ is the number of vertices. Making use of the algorithm on trees, we present a $$k$$-approximation algorithm for the minimum (cardinality) $$\mathrm{CVCP}_k$$ problem under the assumption that the graph has girth at least $$k$$. An example is given showing that performance ratio $$k$$ is asymptotically tight for our algorithm.

##### MSC:
 05C40 Connectivity 05C38 Paths and cycles 05C05 Trees 68W25 Approximation algorithms 68Q25 Analysis of algorithms and problem complexity
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