Approximation Algorithm for the Minimum Connected k -Path Vertex Cover Problem

A vertex subset \(C\) of a connected graph \(G\) is called a connected \(k\)-path vertex cover (\(CVCP_k\)) if every path of length \(k-1\) contains at least one vertex from \(C\), and the subgraph of \(G\) induced by \(C\) is connected. This concept has its background in the field of security and supervisory control. A variation, called \(CVCC_k\), asks every connected subgraph on \(k\) vertices contains at least one vertex from \(C\). The \(MCVCP_k\) (resp. \(MCVCC_k\)) problem is to find a \(CVCP_k\) (resp. \(CVCC_k\)) with the minimum cardinality. In this paper, we give a \(k\)-approximation algorithm for \(MCVCP_k\) under the assumption that the graph has girth at least \(k\). Similar algorithm on \(MCVCC_k\) also yields approximation ratio \(k\), which is valid for any connected graph (without additional conditions).