Variant of an approximation algorithm for vertex cover
Consider a star consisting of a center $x$ connected to the vertices $y_1,\ldots,y_n$.
Your modified algorithm could act as follows:


Choose the edge $(x,y_1)$; add $x,y_1$ to the vertex cover; remove all edges incident to $y_1$.
Choose the edge $(x,y_2)$; add $x,y_2$ to the vertex cover; remove all edges incident to $y_2$.
...


I'll let you take it from here.