Let \(E\) be a pluripolar set in \(\mathbb{C}^N\) and \(D\) be its neighborhood. Two types of pluripolar hulls of \(E\) relative to \(D\) are considered: \[ E^*_D:= \cap\bigl \{z\in D;u(z)= -\infty\bigr\}\quad\text{and}\quad E^-_D:= \cap\bigl\{z \in D;u(z)= -\infty\bigr\} \] where the intersections are taken over all plurisubharmonic functions in \(D\) that are \(-\infty\) on \(E\) in the first case and over such negative functions in the second one. The authors establish two criteria for a point to belong to \(E^-_D\) and investigate the constructions of these hulls and the relations between them. In the last section they study the pluripolar hull of the set \(\{(z,w)\in \mathbb{C}^2;\;w=z^\alpha, \;z\neq 0\}\) with the irrational number \(\alpha>0\) and crack one of the Sadullaev's problems.