Treewidth and Pathwidth parameterized by the vertex cover number

After the number of vertices, Vertex Cover Number is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex Cover Number. Here we consider the treewidth and pathwidth problems parameterized by k, the size of a minimum vertex cover of the input graph. We show that the pathwidth and treewidth can be computed in O∗(3k) time. This complements recent polynomial kernel results for treewidth and pathwidth parameterized by the Vertex Cover Number.