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A vertex subset C of a connected graph G is called a connected k -path vertex cover ( C V C P 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) C V C P k problem. We first show that the minimum weight C V C P 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) C V C P 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. [ABSTRACT FROM AUTHOR]