Vertex cover problem is a classical NP complete problem,which has many applications in areas of sorting,computer network,etc.In recent years,many researchers begin to explore its generalized form,namely vertex cover P_k problem,which requires to find a subset of vertices such that the induced subgraphs by the rest vertices of the topological structure do not include any P_k path,where P_k is the path on k vertices.This paper firstly introduces the application background of vertex cover P_k problem,then summarizes current researches in approximation algorithms,exact algorithms and parameter algorithms,and analyzes some commonly used methods and techniques.On this basis,we point out the direction for further research on the vertex cover P_k problem and its related issues.