On the parameterized vertex cover problem for graphs with perfect matching

A vertex cover of an n-vertex graph with perfect matching contains at least n/2 vertices.In this paper,we study the parameterized complexity of the problem vc-pm^＊that decides if a given graph with perfect matching has a vertex cover of size bounded by n/2＋k.We first present an algorithm of running time O^＊（4k）for a variation of the vertex cover problem on Konig graphs with perfect matching.This algorithm combined with the iterative compression technique leads to an algorithm of running time O^＊（9k）for the problem vc-pm^＊.Our result improves the previous best algorithm of running time O^＊（15k）for the vc-pm^＊problem,which reduces the problem to the almost 2-sat problem and solves the latter by Razgon and O’Sullivan’s recent algorithm.