Removability of product sets for Sobolev functions in the plane

We study conditions on closed sets C, F subset of R making the product CxF removable or non-removable for W-1,W-p. The main results show that the Hausdorff-dimension of the smaller dimensional component C determines a critical exponent above which the product is removable for some positive measure sets F, but below which the product is not removable for another collection of positive measure totally disconnected sets F. Moreover, if the set C is Ahlfors-regular, the above removability holds for any totally disconnected F.